Mathieu group M22
The claims on this page come from https://brauer.maths.qmul.ac.uk/Atlas/v3/scripts/group2.php?id=167&subpage=11.
Claim
\[
\mathrm{M}_{22} = \langle a, b \mid a^2 = b^4 = (ab)^{11} = (ab^2)^5 = [a,bab]^3 = (ababab^{−1})^5 = 1 \rangle
\]
with
\[
|\mathrm{M}_{22}| = 443,520.
\]
On this page, we verify that the above claimed presentation of the Mathieu group M22 defines a group of order \(443,520\).
The code
In libsemigroups_pybind11, the following script constructs the presentation for M11 and runs the Todd-Coxeter algorithm.
Code
from libsemigroups_pybind11 import (
Presentation,
ToddCoxeter,
congruence_kind,
presentation,
)
from libsemigroups_pybind11.words import parse_relations
# Setup the presentation object with the empty word and inverses, so it can represent a group
p = Presentation("abAB")
p.contains_empty_word(True)
presentation.add_inverse_rules(p, "ABab")
# Add the defining relations
presentation.add_rule(p, parse_relations("a^2"), "")
presentation.add_rule(p, parse_relations("b^4"), "")
presentation.add_rule(p, parse_relations("(ab)^11"), "")
presentation.add_rule(p, parse_relations("(ab^2)^5"), "")
presentation.add_rule(p, parse_relations("(a,bab)^3"), "")
presentation.add_rule(p, parse_relations("(ababaB)^5"), "")
# Run the Todd-Coxeter algorithm
tc = ToddCoxeter(congruence_kind.twosided, p)
tc.run()
print(f"The size of the group is {tc.number_of_classes()}")
The output
Running the above script produces the following output:
Output from the Python script
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: running for approx. 30.000s
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: RUN 0 START (strategy() = hlt)
#0: ToddCoxeter: |A| = 4, |R| = 10, |u| + |v| ∈ [2, 30], ∑(|u| + |v|) = 105
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: HLT 0.0 START
#0: ToddCoxeter: HLT 0.0.0 | active | killed | defined
#0: ToddCoxeter: nodes | 1 | 0 | 1
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 0 | 4 | 0.0%
#0: ToddCoxeter: time | run 0 = 19µs | all runs = 19µs | elapsed = 89µs
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: HLT 0.0 STOP
#0: ToddCoxeter: HLT 0.0.1 | active | killed | defined
#0: ToddCoxeter: nodes | 5,000,018 | 1,632,158 | 6,632,176
#0: ToddCoxeter: diff 0.0.0 | +5,000,017 | +1,632,158 | +6,632,175
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 5,679,076 | 14,320,996 | 28.4%
#0: ToddCoxeter: diff 0.0.0 | +5,679,076 | +14,320,992 | +28.4%
#0: ToddCoxeter: phase 0.0 = 547ms | run 0 = 547ms | all runs = 547ms | elapsed = 547ms
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: LOOKAHEAD 0.1 START (lookahead_extent() = partial, lookahead_style() = hlt)
#0: ToddCoxeter: LOOKAHEAD 0.1.0 | active | killed | defined
#0: ToddCoxeter: nodes | 5,000,018 | 1,632,158 | 6,632,176
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 5,679,076 | 14,320,996 | 28.4%
#0: ToddCoxeter: time | run 0 = 547ms | all runs = 547ms | elapsed = 547ms
#0: ToddCoxeter: because a >= n
#0: ToddCoxeter: where: a = number_of_nodes_active() = 5,000,018
#0: ToddCoxeter: n = lookahead_next() = 5,000,000
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: LOOKAHEAD 0.1.1 | active | killed | defined
#1: ToddCoxeter: nodes | 2,644,921 | 3,987,547 | 6,632,176
#1: ToddCoxeter: diff 0.1.0 | -2,355,097 | +2,355,389 | +0
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 7,553,691 | 3,025,993 | 71.4%
#1: ToddCoxeter: diff 0.1.0 | +1,874,615 | -11,295,003 | +43.0%
#1: ToddCoxeter: phase 0.1 = 452ms | run 0 = 1.000s | all runs = 1.000s | elapsed = 1.001s
#1: ToddCoxeter: lookahead progress | ~60.5%
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: LOOKAHEAD 0.1 STOP
#0: ToddCoxeter: LOOKAHEAD 0.1.2 | active | killed | defined
#0: ToddCoxeter: nodes | 2,423,713 | 4,208,463 | 6,632,176
#0: ToddCoxeter: diff 0.1.1 | -221,208 | +220,916 | +0
#0: ToddCoxeter: diff 0.1.0 | -2,576,305 | +2,576,305 | +0
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 7,815,443 | 1,879,409 | 80.6%
#0: ToddCoxeter: diff 0.1.1 | +261,752 | -1,146,584 | +9.2%
#0: ToddCoxeter: diff 0.1.0 | +2,136,367 | -12,441,587 | +52.2%
#0: ToddCoxeter: phase 0.1 = 500ms | run 0 = 1.049s | all runs = 1.049s | elapsed = 1.049s
#0: ToddCoxeter: lookahead_next() is now max(f x a = 4,847,426, m = 10,000) (-152,574)
#0: ToddCoxeter: because f x a < n
#0: ToddCoxeter: where: a = number_of_nodes_active() = 2,423,713
#0: ToddCoxeter: f = lookahead_growth_factor() = 2
#0: ToddCoxeter: m = lookahead_min() = 10,000
#0: ToddCoxeter: n = lookahead_next() = 5,000,000
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: HLT 0.2 START
#0: ToddCoxeter: HLT 0.2.0 | active | killed | defined
#0: ToddCoxeter: nodes | 2,423,713 | 4,208,463 | 6,632,176
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 7,815,443 | 1,879,409 | 80.6%
#0: ToddCoxeter: time | run 0 = 1.049s | all runs = 1.049s | elapsed = 1.049s
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: HLT 0.2 STOP
#0: ToddCoxeter: HLT 0.2.1 | active | killed | defined
#0: ToddCoxeter: nodes | 4,847,474 | 4,320,458 | 9,167,932
#0: ToddCoxeter: diff 0.2.0 | +2,423,761 | +111,995 | +2,535,756
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 10,422,837 | 8,967,059 | 53.8%
#0: ToddCoxeter: diff 0.2.0 | +2,607,394 | +7,087,650 | -26.9%
#0: ToddCoxeter: phase 0.2 = 89ms | run 0 = 1.138s | all runs = 1.138s | elapsed = 1.138s
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: LOOKAHEAD 0.3 START (lookahead_extent() = partial, lookahead_style() = hlt)
#0: ToddCoxeter: LOOKAHEAD 0.3.0 | active | killed | defined
#0: ToddCoxeter: nodes | 4,847,474 | 4,320,458 | 9,167,932
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 10,422,837 | 8,967,059 | 53.8%
#0: ToddCoxeter: time | run 0 = 1.138s | all runs = 1.138s | elapsed = 1.138s
#0: ToddCoxeter: because a >= n
#0: ToddCoxeter: where: a = number_of_nodes_active() = 4,847,474
#0: ToddCoxeter: n = lookahead_next() = 4,847,426
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: LOOKAHEAD 0.3 STOP
#0: ToddCoxeter: LOOKAHEAD 0.3.1 | active | killed | defined
#0: ToddCoxeter: nodes | 3,512,400 | 5,655,532 | 9,167,932
#0: ToddCoxeter: diff 0.3.0 | -1,335,074 | +1,335,074 | +0
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 11,491,787 | 2,557,813 | 81.8%
#0: ToddCoxeter: diff 0.3.0 | +1,068,950 | -6,409,246 | +28.0%
#0: ToddCoxeter: phase 0.3 = 718ms | run 0 = 1.856s | all runs = 1.856s | elapsed = 1.857s
#0: ToddCoxeter: lookahead_next() is now 4,847,426 (+0)
#0: ToddCoxeter: because:
#0: ToddCoxeter: 1. n <= f x a = 7,024,800
#0: ToddCoxeter: 2. a <= n
#0: ToddCoxeter: 3. l >= (l + a) / t = 1,211,868
#0: ToddCoxeter: where: a = number_of_nodes_active() = 3,512,400
#0: ToddCoxeter: f = lookahead_growth_factor() = 2
#0: ToddCoxeter: l = nodes killed in lookahead = 1,335,074
#0: ToddCoxeter: n = lookahead_next() = 4,847,426
#0: ToddCoxeter: t = lookahead_growth_threshold() = 4
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: HLT 0.4 START
#0: ToddCoxeter: HLT 0.4.0 | active | killed | defined
#0: ToddCoxeter: nodes | 3,512,400 | 5,655,532 | 9,167,932
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 11,491,787 | 2,557,813 | 81.8%
#0: ToddCoxeter: time | run 0 = 1.857s | all runs = 1.857s | elapsed = 1.857s
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: HLT 0.4 STOP
#0: ToddCoxeter: HLT 0.4.1 | active | killed | defined
#0: ToddCoxeter: nodes | 4,847,449 | 5,703,629 | 10,551,078
#0: ToddCoxeter: diff 0.4.0 | +1,335,049 | +48,097 | +1,383,146
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 12,934,269 | 6,455,527 | 66.7%
#0: ToddCoxeter: diff 0.4.0 | +1,442,482 | +3,897,714 | -15.1%
#0: ToddCoxeter: phase 0.4 = 67ms | run 0 = 1.924s | all runs = 1.924s | elapsed = 1.924s
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: LOOKAHEAD 0.5 START (lookahead_extent() = partial, lookahead_style() = hlt)
#0: ToddCoxeter: LOOKAHEAD 0.5.0 | active | killed | defined
#0: ToddCoxeter: nodes | 4,847,449 | 5,703,629 | 10,551,078
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 12,934,269 | 6,455,527 | 66.7%
#0: ToddCoxeter: time | run 0 = 1.924s | all runs = 1.924s | elapsed = 1.924s
#0: ToddCoxeter: because a >= n
#0: ToddCoxeter: where: a = number_of_nodes_active() = 4,847,449
#0: ToddCoxeter: n = lookahead_next() = 4,847,426
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: LOOKAHEAD 0.5.1 | active | killed | defined
#1: ToddCoxeter: nodes | 4,820,487 | 5,730,591 | 10,551,078
#1: ToddCoxeter: diff 0.5.0 | -26,962 | +26,962 | +0
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 12,908,450 | 6,373,498 | 66.9%
#1: ToddCoxeter: diff 0.5.0 | -25,819 | -82,029 | +0.2%
#1: ToddCoxeter: phase 0.5 = 77ms | run 0 = 2.001s | all runs = 2.001s | elapsed = 2.001s
#1: ToddCoxeter: lookahead progress | ~11.0%
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: LOOKAHEAD 0.5 STOP
#0: ToddCoxeter: LOOKAHEAD 0.5.2 | active | killed | defined
#0: ToddCoxeter: nodes | 4,134,106 | 6,416,972 | 10,551,078
#0: ToddCoxeter: diff 0.5.1 | -686,381 | +686,381 | +0
#0: ToddCoxeter: diff 0.5.0 | -713,343 | +713,343 | +0
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 13,557,552 | 2,978,872 | 82.0%
#0: ToddCoxeter: diff 0.5.1 | +649,102 | -3,394,626 | +15.0%
#0: ToddCoxeter: diff 0.5.0 | +623,283 | -3,476,655 | +15.3%
#0: ToddCoxeter: phase 0.5 = 856ms | run 0 = 2.780s | all runs = 2.780s | elapsed = 2.780s
#0: ToddCoxeter: lookahead_next() is now n x f = 9,694,852 (+4,847,426)
#0: ToddCoxeter: because: l < (l + a) / t = 1,211,862
#0: ToddCoxeter: where: a = number_of_nodes_active() = 4,134,106
#0: ToddCoxeter: f = lookahead_growth_factor() = 2
#0: ToddCoxeter: l = nodes killed in lookahead = 713,343
#0: ToddCoxeter: n = lookahead_next() = 4,847,426
#0: ToddCoxeter: t = lookahead_growth_threshold() = 4
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: HLT 0.6 START
#0: ToddCoxeter: HLT 0.6.0 | active | killed | defined
#0: ToddCoxeter: nodes | 4,134,106 | 6,416,972 | 10,551,078
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 13,557,552 | 2,978,872 | 82.0%
#0: ToddCoxeter: time | run 0 = 2.780s | all runs = 2.780s | elapsed = 2.780s
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: HLT 0.6.1 | active | killed | defined
#1: ToddCoxeter: nodes | 8,388,608 | 6,594,998 | 14,983,606
#1: ToddCoxeter: diff 0.6.0 | +4,254,502 | +178,026 | +4,432,528
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 18,160,579 | 15,393,853 | 54.1%
#1: ToddCoxeter: diff 0.6.0 | +4,603,027 | +12,414,981 | -27.9%
#1: ToddCoxeter: phase 0.6 = 221ms | run 0 = 3.001s | all runs = 3.001s | elapsed = 3.001s
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: HLT 0.6 STOP
#0: ToddCoxeter: HLT 0.6.2 | active | killed | defined
#0: ToddCoxeter: nodes | 9,694,873 | 6,657,105 | 16,351,978
#0: ToddCoxeter: diff 0.6.1 | +1,306,265 | +62,107 | +1,368,372
#0: ToddCoxeter: diff 0.6.0 | +5,560,767 | +240,133 | +5,800,900
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 19,575,370 | 19,204,122 | 50.5%
#0: ToddCoxeter: diff 0.6.1 | +1,414,791 | +3,810,269 | -3.6%
#0: ToddCoxeter: diff 0.6.0 | +6,017,818 | +16,225,250 | -31.5%
#0: ToddCoxeter: phase 0.6 = 520ms | run 0 = 3.301s | all runs = 3.301s | elapsed = 3.301s
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: LOOKAHEAD 0.7 START (lookahead_extent() = partial, lookahead_style() = hlt)
#0: ToddCoxeter: LOOKAHEAD 0.7.0 | active | killed | defined
#0: ToddCoxeter: nodes | 9,694,873 | 6,657,105 | 16,351,978
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 19,575,370 | 19,204,122 | 50.5%
#0: ToddCoxeter: time | run 0 = 3.301s | all runs = 3.301s | elapsed = 3.301s
#0: ToddCoxeter: because a >= n
#0: ToddCoxeter: where: a = number_of_nodes_active() = 9,694,873
#0: ToddCoxeter: n = lookahead_next() = 9,694,852
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: LOOKAHEAD 0.7.1 | active | killed | defined
#1: ToddCoxeter: nodes | 9,431,978 | 6,920,002 | 16,351,978
#1: ToddCoxeter: diff 0.7.0 | -262,895 | +262,897 | +0
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 19,360,399 | 18,367,513 | 51.3%
#1: ToddCoxeter: diff 0.7.0 | -214,971 | -836,609 | +0.8%
#1: ToddCoxeter: phase 0.7 = 700ms | run 0 = 4.001s | all runs = 4.001s | elapsed = 4.001s
#1: ToddCoxeter: lookahead progress | ~41.5%
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: LOOKAHEAD 0.7 STOP
#0: ToddCoxeter: LOOKAHEAD 0.7.2 | active | killed | defined
#0: ToddCoxeter: nodes | 6,784,030 | 9,567,948 | 16,351,978
#0: ToddCoxeter: diff 0.7.1 | -2,647,948 | +2,647,946 | +0
#0: ToddCoxeter: diff 0.7.0 | -2,910,843 | +2,910,843 | +0
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 22,154,799 | 4,981,321 | 81.6%
#0: ToddCoxeter: diff 0.7.1 | +2,794,400 | -13,386,192 | +30.3%
#0: ToddCoxeter: diff 0.7.0 | +2,579,429 | -14,222,801 | +31.2%
#0: ToddCoxeter: phase 0.7 = 1.445s | run 0 = 4.746s | all runs = 4.746s | elapsed = 4.746s
#0: ToddCoxeter: lookahead_next() is now 9,694,852 (+0)
#0: ToddCoxeter: because:
#0: ToddCoxeter: 1. n <= f x a = 13,568,060
#0: ToddCoxeter: 2. a <= n
#0: ToddCoxeter: 3. l >= (l + a) / t = 2,423,718
#0: ToddCoxeter: where: a = number_of_nodes_active() = 6,784,030
#0: ToddCoxeter: f = lookahead_growth_factor() = 2
#0: ToddCoxeter: l = nodes killed in lookahead = 2,910,843
#0: ToddCoxeter: n = lookahead_next() = 9,694,852
#0: ToddCoxeter: t = lookahead_growth_threshold() = 4
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: HLT 0.8 START
#0: ToddCoxeter: HLT 0.8.0 | active | killed | defined
#0: ToddCoxeter: nodes | 6,784,030 | 9,567,948 | 16,351,978
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 22,154,799 | 4,981,321 | 81.6%
#0: ToddCoxeter: time | run 0 = 4.746s | all runs = 4.746s | elapsed = 4.746s
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: HLT 0.8 STOP
#0: ToddCoxeter: HLT 0.8.1 | active | killed | defined
#0: ToddCoxeter: nodes | 9,694,854 | 9,675,719 | 19,370,573
#0: ToddCoxeter: diff 0.8.0 | +2,910,824 | +107,771 | +3,018,595
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 25,297,258 | 13,482,158 | 65.2%
#0: ToddCoxeter: diff 0.8.0 | +3,142,459 | +8,500,837 | -16.4%
#0: ToddCoxeter: phase 0.8 = 130ms | run 0 = 4.876s | all runs = 4.876s | elapsed = 4.876s
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: LOOKAHEAD 0.9 START (lookahead_extent() = partial, lookahead_style() = hlt)
#0: ToddCoxeter: LOOKAHEAD 0.9.0 | active | killed | defined
#0: ToddCoxeter: nodes | 9,694,854 | 9,675,719 | 19,370,573
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 25,297,258 | 13,482,158 | 65.2%
#0: ToddCoxeter: time | run 0 = 4.876s | all runs = 4.876s | elapsed = 4.876s
#0: ToddCoxeter: because a >= n
#0: ToddCoxeter: where: a = number_of_nodes_active() = 9,694,854
#0: ToddCoxeter: n = lookahead_next() = 9,694,852
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: LOOKAHEAD 0.9.1 | active | killed | defined
#1: ToddCoxeter: nodes | 9,648,811 | 9,721,762 | 19,370,573
#1: ToddCoxeter: diff 0.9.0 | -46,043 | +46,043 | +0
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 25,256,260 | 13,338,984 | 65.4%
#1: ToddCoxeter: diff 0.9.0 | -40,998 | -143,174 | +0.2%
#1: ToddCoxeter: phase 0.9 = 125ms | run 0 = 5.001s | all runs = 5.001s | elapsed = 5.002s
#1: ToddCoxeter: lookahead progress | ~9.5%
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: LOOKAHEAD 0.9.2 | active | killed | defined
#1: ToddCoxeter: nodes | 9,403,427 | 9,967,160 | 19,370,573
#1: ToddCoxeter: diff 0.9.1 | -245,384 | +245,398 | +0
#1: ToddCoxeter: diff 0.9.0 | -291,427 | +291,441 | +0
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 25,010,253 | 12,603,455 | 66.5%
#1: ToddCoxeter: diff 0.9.1 | -246,007 | -735,529 | +1.1%
#1: ToddCoxeter: diff 0.9.0 | -287,005 | -878,703 | +1.3%
#1: ToddCoxeter: phase 0.9 = 1.126s | run 0 = 6.002s | all runs = 6.002s | elapsed = 6.002s
#1: ToddCoxeter: lookahead progress | ~63.4%
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: LOOKAHEAD 0.9 STOP
#0: ToddCoxeter: LOOKAHEAD 0.9.3 | active | killed | defined
#0: ToddCoxeter: nodes | 8,086,523 | 11,284,050 | 19,370,573
#0: ToddCoxeter: diff 0.9.2 | -1,316,904 | +1,316,890 | +0
#0: ToddCoxeter: diff 0.9.0 | -1,608,331 | +1,608,331 | +0
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 26,555,088 | 5,791,004 | 82.1%
#0: ToddCoxeter: diff 0.9.2 | +1,544,835 | -6,812,451 | +15.6%
#0: ToddCoxeter: diff 0.9.0 | +1,257,830 | -7,691,154 | +16.9%
#0: ToddCoxeter: phase 0.9 = 1.690s | run 0 = 6.566s | all runs = 6.566s | elapsed = 6.566s
#0: ToddCoxeter: lookahead_next() is now n x f = 19,389,704 (+9,694,852)
#0: ToddCoxeter: because: l < (l + a) / t = 2,423,713
#0: ToddCoxeter: where: a = number_of_nodes_active() = 8,086,523
#0: ToddCoxeter: f = lookahead_growth_factor() = 2
#0: ToddCoxeter: l = nodes killed in lookahead = 1,608,331
#0: ToddCoxeter: n = lookahead_next() = 9,694,852
#0: ToddCoxeter: t = lookahead_growth_threshold() = 4
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: HLT 0.10 START
#0: ToddCoxeter: HLT 0.10.0 | active | killed | defined
#0: ToddCoxeter: nodes | 8,086,523 | 11,284,050 | 19,370,573
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 26,555,088 | 5,791,004 | 82.1%
#0: ToddCoxeter: time | run 0 = 6.566s | all runs = 6.566s | elapsed = 6.566s
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: HLT 0.10.1 | active | killed | defined
#1: ToddCoxeter: nodes | 16,777,216 | 11,642,699 | 28,419,915
#1: ToddCoxeter: diff 0.10.0 | +8,690,693 | +358,649 | +9,049,342
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 35,954,563 | 31,154,301 | 53.6%
#1: ToddCoxeter: diff 0.10.0 | +9,399,475 | +25,363,297 | -28.5%
#1: ToddCoxeter: phase 0.10 = 436ms | run 0 = 7.002s | all runs = 7.002s | elapsed = 7.002s
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: HLT 0.10 STOP
#0: ToddCoxeter: HLT 0.10.2 | active | killed | defined
#0: ToddCoxeter: nodes | 19,389,750 | 11,773,835 | 31,163,585
#0: ToddCoxeter: diff 0.10.1 | +2,612,534 | +131,136 | +2,743,670
#0: ToddCoxeter: diff 0.10.0 | +11,303,227 | +489,785 | +11,793,012
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 38,786,740 | 38,772,260 | 50.0%
#0: ToddCoxeter: diff 0.10.1 | +2,832,177 | +7,617,959 | -3.6%
#0: ToddCoxeter: diff 0.10.0 | +12,231,652 | +32,981,256 | -32.1%
#0: ToddCoxeter: phase 0.10 = 1.051s | run 0 = 7.617s | all runs = 7.617s | elapsed = 7.617s
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: LOOKAHEAD 0.11 START (lookahead_extent() = partial, lookahead_style() = hlt)
#0: ToddCoxeter: LOOKAHEAD 0.11.0 | active | killed | defined
#0: ToddCoxeter: nodes | 19,389,750 | 11,773,835 | 31,163,585
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 38,786,740 | 38,772,260 | 50.0%
#0: ToddCoxeter: time | run 0 = 7.617s | all runs = 7.617s | elapsed = 7.617s
#0: ToddCoxeter: because a >= n
#0: ToddCoxeter: where: a = number_of_nodes_active() = 19,389,750
#0: ToddCoxeter: n = lookahead_next() = 19,389,704
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: LOOKAHEAD 0.11.1 | active | killed | defined
#1: ToddCoxeter: nodes | 19,119,246 | 12,044,357 | 31,163,585
#1: ToddCoxeter: diff 0.11.0 | -270,504 | +270,522 | +0
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 38,564,145 | 37,912,839 | 50.4%
#1: ToddCoxeter: diff 0.11.0 | -222,595 | -859,421 | +0.4%
#1: ToddCoxeter: phase 0.11 = 385ms | run 0 = 8.002s | all runs = 8.002s | elapsed = 8.002s
#1: ToddCoxeter: lookahead progress | ~12.6%
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: LOOKAHEAD 0.11.2 | active | killed | defined
#1: ToddCoxeter: nodes | 18,894,507 | 12,269,078 | 31,163,585
#1: ToddCoxeter: diff 0.11.1 | -224,739 | +224,721 | +0
#1: ToddCoxeter: diff 0.11.0 | -495,243 | +495,243 | +0
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 38,371,598 | 37,206,430 | 50.8%
#1: ToddCoxeter: diff 0.11.1 | -192,547 | -706,409 | +0.3%
#1: ToddCoxeter: diff 0.11.0 | -415,142 | -1,565,830 | +0.8%
#1: ToddCoxeter: phase 0.11 = 1.385s | run 0 = 9.003s | all runs = 9.003s | elapsed = 9.003s
#1: ToddCoxeter: lookahead progress | ~38.8%
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: LOOKAHEAD 0.11.3 | active | killed | defined
#1: ToddCoxeter: nodes | 15,851,580 | 15,312,085 | 31,163,585
#1: ToddCoxeter: diff 0.11.2 | -3,042,927 | +3,043,007 | +0
#1: ToddCoxeter: diff 0.11.0 | -3,538,170 | +3,538,250 | +0
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 41,333,919 | 22,072,401 | 65.2%
#1: ToddCoxeter: diff 0.11.2 | +2,962,321 | -15,134,029 | +14.4%
#1: ToddCoxeter: diff 0.11.0 | +2,547,179 | -16,699,859 | +15.2%
#1: ToddCoxeter: phase 0.11 = 2.386s | run 0 = 10.003s | all runs = 10.003s | elapsed = 10.003s
#1: ToddCoxeter: lookahead progress | ~62.6%
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: LOOKAHEAD 0.11 STOP
#0: ToddCoxeter: LOOKAHEAD 0.11.4 | active | killed | defined
#0: ToddCoxeter: nodes | 13,427,510 | 17,736,075 | 31,163,585
#0: ToddCoxeter: diff 0.11.3 | -2,424,070 | +2,423,990 | +0
#0: ToddCoxeter: diff 0.11.0 | -5,962,240 | +5,962,240 | +0
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 43,882,513 | 9,827,527 | 81.7%
#0: ToddCoxeter: diff 0.11.3 | +2,548,594 | -12,244,874 | +16.5%
#0: ToddCoxeter: diff 0.11.0 | +5,095,773 | -28,944,733 | +31.7%
#0: ToddCoxeter: phase 0.11 = 2.893s | run 0 = 10.511s | all runs = 10.511s | elapsed = 10.511s
#0: ToddCoxeter: lookahead_next() is now 19,389,704 (+0)
#0: ToddCoxeter: because:
#0: ToddCoxeter: 1. n <= f x a = 26,855,020
#0: ToddCoxeter: 2. a <= n
#0: ToddCoxeter: 3. l >= (l + a) / t = 4,847,437
#0: ToddCoxeter: where: a = number_of_nodes_active() = 13,427,510
#0: ToddCoxeter: f = lookahead_growth_factor() = 2
#0: ToddCoxeter: l = nodes killed in lookahead = 5,962,240
#0: ToddCoxeter: n = lookahead_next() = 19,389,704
#0: ToddCoxeter: t = lookahead_growth_threshold() = 4
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: HLT 0.12 START
#0: ToddCoxeter: HLT 0.12.0 | active | killed | defined
#0: ToddCoxeter: nodes | 13,427,510 | 17,736,075 | 31,163,585
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 43,882,513 | 9,827,527 | 81.7%
#0: ToddCoxeter: time | run 0 = 10.511s | all runs = 10.511s | elapsed = 10.511s
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: HLT 0.12 STOP
#0: ToddCoxeter: HLT 0.12.1 | active | killed | defined
#0: ToddCoxeter: nodes | 19,389,776 | 17,959,888 | 37,349,664
#0: ToddCoxeter: diff 0.12.0 | +5,962,266 | +223,813 | +6,186,079
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 50,324,580 | 27,234,524 | 64.9%
#0: ToddCoxeter: diff 0.12.0 | +6,442,067 | +17,406,997 | -16.8%
#0: ToddCoxeter: phase 0.12 = 267ms | run 0 = 10.778s | all runs = 10.778s | elapsed = 10.779s
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: LOOKAHEAD 0.13 START (lookahead_extent() = partial, lookahead_style() = hlt)
#0: ToddCoxeter: LOOKAHEAD 0.13.0 | active | killed | defined
#0: ToddCoxeter: nodes | 19,389,776 | 17,959,888 | 37,349,664
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 50,324,580 | 27,234,524 | 64.9%
#0: ToddCoxeter: time | run 0 = 10.778s | all runs = 10.778s | elapsed = 10.779s
#0: ToddCoxeter: because a >= n
#0: ToddCoxeter: where: a = number_of_nodes_active() = 19,389,776
#0: ToddCoxeter: n = lookahead_next() = 19,389,704
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: LOOKAHEAD 0.13.1 | active | killed | defined
#1: ToddCoxeter: nodes | 19,314,200 | 18,035,464 | 37,349,664
#1: ToddCoxeter: diff 0.13.0 | -75,576 | +75,576 | +0
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 50,245,486 | 27,011,314 | 65.0%
#1: ToddCoxeter: diff 0.13.0 | -79,094 | -223,210 | +0.2%
#1: ToddCoxeter: phase 0.13 = 224ms | run 0 = 11.003s | all runs = 11.003s | elapsed = 11.003s
#1: ToddCoxeter: lookahead progress | ~9.0%
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: LOOKAHEAD 0.13.2 | active | killed | defined
#1: ToddCoxeter: nodes | 19,102,330 | 18,247,350 | 37,349,664
#1: ToddCoxeter: diff 0.13.1 | -211,870 | +211,886 | +0
#1: ToddCoxeter: diff 0.13.0 | -287,446 | +287,462 | +0
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 50,032,914 | 26,376,406 | 65.5%
#1: ToddCoxeter: diff 0.13.1 | -212,572 | -634,908 | +0.4%
#1: ToddCoxeter: diff 0.13.0 | -291,666 | -858,118 | +0.6%
#1: ToddCoxeter: phase 0.13 = 1.225s | run 0 = 12.003s | all runs = 12.003s | elapsed = 12.003s
#1: ToddCoxeter: lookahead progress | ~31.3%
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: LOOKAHEAD 0.13.3 | active | killed | defined
#1: ToddCoxeter: nodes | 18,872,056 | 18,477,610 | 37,349,664
#1: ToddCoxeter: diff 0.13.2 | -230,274 | +230,260 | +0
#1: ToddCoxeter: diff 0.13.0 | -517,720 | +517,722 | +0
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 49,788,690 | 25,699,534 | 66.0%
#1: ToddCoxeter: diff 0.13.2 | -244,224 | -676,872 | +0.5%
#1: ToddCoxeter: diff 0.13.0 | -535,890 | -1,534,990 | +1.1%
#1: ToddCoxeter: phase 0.13 = 2.225s | run 0 = 13.004s | all runs = 13.004s | elapsed = 13.004s
#1: ToddCoxeter: lookahead progress | ~58.4%
#0: ToddCoxeter: large collapse, number of coincidences 100,000 >= 100,000 = large_collapse()!
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: LOOKAHEAD 0.13.4 | active | killed | defined
#1: ToddCoxeter: nodes | 12,447,893 | 24,901,915 | 37,349,664
#1: ToddCoxeter: diff 0.13.3 | -6,424,163 | +6,424,305 | +0
#1: ToddCoxeter: diff 0.13.0 | -6,941,883 | +6,942,027 | +0
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 0 | 49,791,572 | 0.0%
#1: ToddCoxeter: diff 0.13.3 | -49,788,690 | +24,092,038 | -66.0%
#1: ToddCoxeter: diff 0.13.0 | -50,324,580 | +22,557,048 | -64.9%
#1: ToddCoxeter: phase 0.13 = 3.225s | run 0 = 14.004s | all runs = 14.004s | elapsed = 14.004s
#1: ToddCoxeter: lookahead progress | ~66.6%
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: LOOKAHEAD 0.13.5 | active | killed | defined
#1: ToddCoxeter: nodes | 5,024,869 | 32,324,910 | 37,349,664
#1: ToddCoxeter: diff 0.13.4 | -7,423,024 | +7,422,995 | +0
#1: ToddCoxeter: diff 0.13.0 | -14,364,907 | +14,365,022 | +0
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 0 | 20,099,476 | 0.0%
#1: ToddCoxeter: diff 0.13.4 | +0 | -29,692,096 | +0.0%
#1: ToddCoxeter: diff 0.13.0 | -50,324,580 | -7,135,048 | -64.9%
#1: ToddCoxeter: phase 0.13 = 4.226s | run 0 = 15.004s | all runs = 15.004s | elapsed = 15.004s
#1: ToddCoxeter: lookahead progress | ~66.6%
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: LOOKAHEAD 0.13 STOP
#0: ToddCoxeter: LOOKAHEAD 0.13.6 | active | killed | defined
#0: ToddCoxeter: nodes | 443,520 | 36,906,144 | 37,349,664
#0: ToddCoxeter: diff 0.13.5 | -4,581,349 | +4,581,234 | +0
#0: ToddCoxeter: diff 0.13.0 | -18,946,256 | +18,946,256 | +0
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 1,774,080 | 0 | 100.0%
#0: ToddCoxeter: diff 0.13.5 | +1,774,080 | -20,099,476 | +100.0%
#0: ToddCoxeter: diff 0.13.0 | -48,550,500 | -27,234,524 | +35.1%
#0: ToddCoxeter: phase 0.13 = 5.080s | run 0 = 15.858s | all runs = 15.858s | elapsed = 15.858s
#0: ToddCoxeter: lookahead_next() is now max(f x a = 887,040, m = 10,000) (-18,502,664)
#0: ToddCoxeter: because f x a < n
#0: ToddCoxeter: where: a = number_of_nodes_active() = 443,520
#0: ToddCoxeter: f = lookahead_growth_factor() = 2
#0: ToddCoxeter: m = lookahead_min() = 10,000
#0: ToddCoxeter: n = lookahead_next() = 19,389,704
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: HLT 0.14 START
#0: ToddCoxeter: HLT 0.14.0 | active | killed | defined
#0: ToddCoxeter: nodes | 443,520 | 36,906,144 | 37,349,664
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 1,774,080 | 0 | 100.0%
#0: ToddCoxeter: time | run 0 = 15.858s | all runs = 15.858s | elapsed = 15.858s
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: HLT 0.14.1 | active | killed | defined
#1: ToddCoxeter: nodes | 443,520 | 36,906,144 | 37,349,664
#1: ToddCoxeter: diff 0.14.0 | +0 | +0 | +0
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 1,774,080 | 0 | 100.0%
#1: ToddCoxeter: diff 0.14.0 | +0 | +0 | +0.0%
#1: ToddCoxeter: phase 0.14 = 146ms | run 0 = 16.004s | all runs = 16.004s | elapsed = 16.005s
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: HLT 0.14 STOP
#0: ToddCoxeter: HLT 0.14.2 | active | killed | defined
#0: ToddCoxeter: nodes | 443,520 | 36,906,144 | 37,349,664
#0: ToddCoxeter: diff 0.14.1 | +0 | +0 | +0
#0: ToddCoxeter: diff 0.14.0 | +0 | +0 | +0
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 1,774,080 | 0 | 100.0%
#0: ToddCoxeter: diff 0.14.1 | +0 | +0 | +0.0%
#0: ToddCoxeter: diff 0.14.0 | +0 | +0 | +0.0%
#0: ToddCoxeter: phase 0.14 = 151ms | run 0 = 16.009s | all runs = 16.009s | elapsed = 16.010s
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: HLT 0.15.3 | active | killed | defined
#0: ToddCoxeter: nodes | 443,520 | 36,906,144 | 37,349,664
#0: ToddCoxeter: diff 0.15.2 | +0 | +0 | +0
#0: ToddCoxeter: diff 0.15.0 | +0 | +0 | +0
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 1,774,080 | 0 | 100.0%
#0: ToddCoxeter: diff 0.15.2 | +0 | +0 | +0.0%
#0: ToddCoxeter: diff 0.15.0 | +0 | +0 | +0.0%
#0: ToddCoxeter: phase 0.15 = 151ms | run 0 = 16.010s | all runs = 16.010s | elapsed = 16.010s
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: RUN 0 STOP (finished)
#0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch
#0: ToddCoxeter: num. phases | 7 | 0 | 8 | 0
#0: ToddCoxeter: time spent in phases | 13.184s (82%) | - (0%) | 2.826s (18%) | - (0%)
#0: ToddCoxeter: phase 0.15 = 160ms | run 0 = 16.019s | all runs = 16.019s | elapsed = 16.019s
The size of the group is 443520
The computed size of the group matches the size of the group provided on the ATLAS: \(443,520\).