Order = 44352000 = 29.32.53.7.11.
Mult = 2.
Out = 2.

Porting notes

Porting incomplete.

Standard generators

Standard generators of HS are a, b where a is in class 2A, b is in class 5A and ab has order 11.

Standard generators of 2.HS are preimages A, B where B has order 5 and AB has order 11.

Standard generators of HS:2 are c, d where c is in class 2C, d is in class 5C and cd has order 30.

Standard generators of 2.HS.2 are preimages C, D where D has order 5.

Black box algorithms

Finding generators

Group Algorithm File
HS Download
HS:2 Download

Checking generators (semi-presentations)

Group Semi-presentation File
HS 〈〈 a, b | o(a) = 2, o(b) = 5, o(ab) = 11, o(ab2) = 10, o(abab2) = 15 〉〉 Download
HS:2 〈〈 c, d | o(c) = 2, o(d) = 5, o(cd) = 30, o([c,d]) = 3 〉〉 Download

Presentations

Group Presentation Link
HS 〈 a, b | a2 = b5 = (ab)11 = (ab2)10 = [a, b]5 = [a, b2]6 = [a, bab]3 = ababab2ab−1ab−2ab−1ab2abab(ab−2)4 = ab(ab2(ab−2)2)2ab2abab2(ab−1ab2)2 = abab(ab2)2ab(ab−1)2ab(ab2)2ababab−2ab−1ab−2 = 1 〉 Details
HS:2 〈 c, d | c2 = d5 = [c, d]3 = [c, d2]4 = ((cd)4cd−2cd−1cd−1cd−2)2 = [c, dcdcd2cd−2(cd2)2cd−1] = [c, dcdcd−2cd−1(cd−2)3] = [c, d−1cd−1cd2cd(cd2)3] = 1 〉 Details

Representations

Representations of HS

Representations of 2.HS

Representations of HS:2

Representations of 2.HS.2

Maximal subgroups

Maximal subgroups of HS

Subgroup Order Index Programs/reps
M22 443 520 100 Program: Standard generators
U3(5):2 252 000 176 Program: Standard generators
U3(5):2 252 000 176 Program: Standard generators
L3(4):21 40 320 1 100 Program: Standard generators
S8 40 320 1 100 Program: Standard generators
24.S6 11 520 3 850 Program: Generators mapping onto standard generators
43:L3(2) 10 752 4 125 Program: Generators mapping onto standard generators
M11 7 920 5 600 Program: Standard generators
M11 7 920 5 600 Program: Standard generators
4.24.S5 7 680 5 775 Program: Generators mapping onto standard generators
2 × A6.22 2 880 15 400 Program: Generators
5:4 × A5 1 200 36 960 Program: Generators mapping onto standard generators

Maximal subgroups of HS:2

Subgroup Order Index Programs/reps
HS 44 352 000 2 Program: Standard generators
M22:2. 887 040 100 Program: Generators
Program: Generators
L3(4):22 80 640 1 100 Program: Generators
S8 × 2 80 640 1 100 Program: Generators
Program: Generators
25.S6 23 040 3 850 Program: Generators
43:(2 × L3(2)) 21 504 4 125 Program: Generators
Program: Generators
21+6.S5 15 360 5 775 Program: Generators
(2 × A6.22).2 5 760 15 400 Program: Generators
51+2:[25] 4 000 22 176 Program: Generators
5:4 × S5 2 400 36 960 Program: Generators

Conjugacy classes

Conjugacy classes of HS

Conjugacy class Centraliser order Power up Class rep(s)
1A 44 352 000 (ababbb)8
2A 7 680 4A 4B 4C 6B 8A 8B 8C 10A 12A 20A 20B (ababbb)4
2B 2 880 6A 10B (ababbbabababbabaababbbabababbababbbabababb)3
3A 360 6A 6B 12A 15A (ababbbabababbabaababbbabababbababbbabababb)2
4A 3 840 12A 20A 20B (ababbbabababbabaababbbabababb)3
4B 256 8A (ababbb)2
4C 64 8B 8C (abaababbbabababb)2
5A 500 10A 20A 20B (aababbbabababb)4
5B 300 10B 15A (abb)2
5C 25 ababbbabababb
6A 36 ababbbabababbabaababbbabababbababbbabababb
6B 24 12A (ababbbabababbabaababbbabababb)2
7A 7 abababb
8A 16 ababbb
8B 16 abaababbbabababb
8C 16 ababbababbbabababbabaababbbabababbababbbabababb
10A 20 20A 20B (aababbbabababb)2
10B 20 abb
11A 11 11B2 ab
11B 11 11A2 (ab)2
12A 12 ababbbabababbabaababbbabababb
15A 15 ababb
20A 20 20B11 aababbbabababb
20B 20 20A11 (aababbbabababb)11

Download words for class representatives.

Conjugacy classes of HS:2

Conjugacy class Centraliser order Power up Class rep(s)
1A 88 704 000
2A 15 360 4A 4B 4C 6B 8A 8B 10A 12A 20A 4D 4E 8C 8D 12B 20C 20D
2B 5 760 6A 10B 4F 20B
3A 720 6A 6B 12A 15A 6C 6D 6E 12B 30A
4A 7 680 12A 20A
4B 512 8A 8C 8D
4C 128 8B
5A 1 000 10A 20A 20C 20D
5B 600 10B 15A 10C 20B 30A
5C 50 10D
6A 72
6B 48 12A 12B
7A 14 14A
8A 32
8B 16
10A 40 20A 20C 20D
10B 40 20B
11A 11
12A 24
15A 30 30A
20A 20
2C 80 640 6C 6D 10C 14A 30A
2D 3 840 6E 10D
4D 640 20C 20D
4E 192 12B
4F 80 20B
6C 720 30A
6D 72
6E 48
8C 64
8D 64
10C 60 30A
10D 10
12B 24
14A 14
20B 20
20C 20 20D3
20D 20 20C3
30A 30

Download words (if any exist) for class representatives.