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G = C8.11C42  order 128 = 27

5th non-split extension by C8 of C42 acting via C42/C2×C4=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C8.11C42, C23.29SD16, (C2×C16)⋊5C4, C8.1(C4⋊C4), (C2×C8).2Q8, (C2×C8).81D4, (C2×C4).10D8, (C2×C4).6Q16, C4.Q8.3C4, C8.C4⋊2C4, C4.9(C2.D8), C2.2(C8.Q8), (C2×C4).93SD16, C8.41(C22⋊C4), C4.5(Q8⋊C4), (C22×C4).189D4, C4.31(D4⋊C4), (C2×M5(2)).11C2, C22.13(C4.Q8), C4.4(C2.C42), (C22×C8).202C22, C22.1(Q8⋊C4), C22.21(D4⋊C4), C2.13(C22.4Q16), (C2×C8).49(C2×C4), (C2×C4.Q8).1C2, (C2×C4).111(C4⋊C4), (C2×C8.C4).3C2, (C2×C4).62(C22⋊C4), SmallGroup(128,115)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C8 — C8.11C42
Chief series
C1 — C2 — C4 — C8 — C2×C8 — C22×C8 — C2×M5(2) — C8.11C42
Lower central
C1 — C2 — C4 — C8 — C8.11C42
Upper central
C1 — C22 — C22×C4 — C22×C8 — C8.11C42
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C22×C8 — C8.11C42

Generators and relations for C8.11C42
 G = < a,b,c | a8=b4=1, c4=a2, bab-1=a3, ac=ca, cbc-1=a3b >

Subgroups: 136 in 68 conjugacy classes, 40 normal (32 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C16, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C4.Q8, C4.Q8, C8.C4, C8.C4, C2×C16, M5(2), C2×C4⋊C4, C22×C8, C2×M4(2), C2×C4.Q8, C2×C8.C4, C2×M5(2), C8.11C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, D8, SD16, Q16, C2.C42, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C22.4Q16, C8.Q8, C8.11C42

Smallest permutation representation of C8.11C42
►On 32 points
Generators in S32
(1 28 5 32 9 20 13 24)(2 29 6 17 10 21 14 25)(3 30 7 18 11 22 15 26)(4 31 8 19 12 23 16 27)

(2 29 10 21)(3 7)(4 19 12 27)(5 13)(6 25 14 17)(8 31 16 23)(11 15)(20 24)(22 30)(28 32)

(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32)
 
G:=sub<Sym(32)| (1,28,5,32,9,20,13,24)(2,29,6,17,10,21,14,25)(3,30,7,18,11,22,15,26)(4,31,8,19,12,23,16,27), (2,29,10,21)(3,7)(4,19,12,27)(5,13)(6,25,14,17)(8,31,16,23)(11,15)(20,24)(22,30)(28,32), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)>;
 
G:=Group( (1,28,5,32,9,20,13,24)(2,29,6,17,10,21,14,25)(3,30,7,18,11,22,15,26)(4,31,8,19,12,23,16,27), (2,29,10,21)(3,7)(4,19,12,27)(5,13)(6,25,14,17)(8,31,16,23)(11,15)(20,24)(22,30)(28,32), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32) );
 
G=PermutationGroup([[(1,28,5,32,9,20,13,24),(2,29,6,17,10,21,14,25),(3,30,7,18,11,22,15,26),(4,31,8,19,12,23,16,27)], [(2,29,10,21),(3,7),(4,19,12,27),(5,13),(6,25,14,17),(8,31,16,23),(11,15),(20,24),(22,30),(28,32)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)]])
 

32 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H8I8J16A···16H
order12222244444444888888888816···16
size1111222222888822224488884···4

32 irreducible representations

dim111111122222224
type+++++-++-
imageC1C2C2C2C4C4C4D4Q8D4D8SD16Q16SD16C8.Q8
kernelC8.11C42C2×C4.Q8C2×C8.C4C2×M5(2)C4.Q8C8.C4C2×C16C2×C8C2×C8C22×C4C2×C4C2×C4C2×C4C23C2
# reps111144421122224

Matrix representation of C8.11C42 ►in GL6(𝔽17)

1600000
0160000
0051200
005500
00516512
00161255
,
400000
16130000
000100
001000
001561212
00129125
,
180000
0160000
0078150
001110015
001511109
0061567

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,5,5,5,16,0,0,12,5,16,12,0,0,0,0,5,5,0,0,0,0,12,5],[4,16,0,0,0,0,0,13,0,0,0,0,0,0,0,1,15,12,0,0,1,0,6,9,0,0,0,0,12,12,0,0,0,0,12,5],[1,0,0,0,0,0,8,16,0,0,0,0,0,0,7,11,15,6,0,0,8,10,11,15,0,0,15,0,10,6,0,0,0,15,9,7] >;
 

C8.11C42 in GAP, Magma, Sage, TeX 

C_8._{11}C_4^2
 
% in TeX
 
G:=Group("C8.11C4^2");
 
// GroupNames label
 
G:=SmallGroup(128,115);
 
// by ID
 
G=gap.SmallGroup(128,115);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,184,1018,1684,102,4037,124]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=b^4=1,c^4=a^2,b*a*b^-1=a^3,a*c=c*a,c*b*c^-1=a^3*b>;
 
// generators/relations
 

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