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G = C8.C42  order 128 = 27

1st non-split extension by C8 of C42 acting via C42/C22=C22

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

Aliases: C8.1C42, M5(2)⋊8C4, C23.33D8, C2.D8⋊9C4, C4.Q8⋊4C4, (C2×C8).5Q8, C8.28(C4⋊C4), (C2×C4).8Q16, (C2×C8).362D4, C4.3(C4.Q8), (C2×C4).21SD16, C2.3(D8⋊2C4), C8.42(C22⋊C4), C4.7(Q8⋊C4), (C22×C4).192D4, C22.3(C2.D8), (C2×M5(2)).14C2, C4.7(C2.C42), (C22×C8).205C22, C23.25D4.9C2, C22.46(D4⋊C4), C2.16(C22.4Q16), (C2×C8).52(C2×C4), (C2×C4.Q8).2C2, (C2×C4).28(C4⋊C4), (C2×C4).230(C22⋊C4), SmallGroup(128,118)

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

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

Generators and relations for C8.C42
 G = < a,b,c | a8=b4=1, c4=a6, bab-1=a-1, cac-1=a5, cbc-1=a3b >

Subgroups: 152 in 70 conjugacy classes, 40 normal (32 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C23, C16, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C4.Q8, C4.Q8, C2.D8, C2×C16, M5(2), M5(2), C2×C4⋊C4, C42⋊C2, C22×C8, C2×C4.Q8, C23.25D4, C2×M5(2), C8.C42
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, D8⋊2C4, C8.C42

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

(1 14 26 23)(2 22 27 13)(3 4 28 29)(5 10 30 19)(6 18 31 9)(7 16 32 25)(8 24 17 15)(11 12 20 21)

(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)| (1,24,13,20,9,32,5,28)(2,17,14,29,10,25,6,21)(3,26,15,22,11,18,7,30)(4,19,16,31,12,27,8,23), (1,14,26,23)(2,22,27,13)(3,4,28,29)(5,10,30,19)(6,18,31,9)(7,16,32,25)(8,24,17,15)(11,12,20,21), (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( (1,24,13,20,9,32,5,28)(2,17,14,29,10,25,6,21)(3,26,15,22,11,18,7,30)(4,19,16,31,12,27,8,23), (1,14,26,23)(2,22,27,13)(3,4,28,29)(5,10,30,19)(6,18,31,9)(7,16,32,25)(8,24,17,15)(11,12,20,21), (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([[(1,24,13,20,9,32,5,28),(2,17,14,29,10,25,6,21),(3,26,15,22,11,18,7,30),(4,19,16,31,12,27,8,23)], [(1,14,26,23),(2,22,27,13),(3,4,28,29),(5,10,30,19),(6,18,31,9),(7,16,32,25),(8,24,17,15),(11,12,20,21)], [(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 2A2B2C2D2E4A4B4C4D4E···4L8A8B8C8D8E8F16A···16H
order12222244444···488888816···16
size11112222228···82222444···4

32 irreducible representations

dim11111112222224
type+++++-+-+
imageC1C2C2C2C4C4C4D4Q8D4SD16Q16D8D8⋊2C4
kernelC8.C42C2×C4.Q8C23.25D4C2×M5(2)C4.Q8C2.D8M5(2)C2×C8C2×C8C22×C4C2×C4C2×C4C23C2
# reps11114442114224

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

1600000
0160000
005500
0012500
00001212
0000512
,
0160000
100000
0000125
000055
0051200
00121200
,
0130000
1300000
000010
000001
0012500
00121200

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,5,12,0,0,0,0,5,5,0,0,0,0,0,0,12,5,0,0,0,0,12,12],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,0,5,12,0,0,0,0,12,12,0,0,12,5,0,0,0,0,5,5,0,0],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,5,12,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

C8.C42 in GAP, Magma, Sage, TeX 

C_8.C_4^2
 
% in TeX
 
G:=Group("C8.C4^2");
 
// GroupNames label
 
G:=SmallGroup(128,118);
 
// by ID
 
G=gap.SmallGroup(128,118);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,723,520,1684,102,4037,124]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=b^4=1,c^4=a^6,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^3*b>;
 
// generators/relations
 

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