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G = C42.2C8  order 128 = 27

2nd non-split extension by C42 of C8 acting via C8/C2=C4

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C42.2C8, C23.25M4(2), (C2×C16)⋊3C4, (C2×C8).1C8, C4.8(C4⋊C8), C4.19(C4×C8), C8.32(C4⋊C4), (C2×C8).26Q8, (C2×C8).182D4, C22.3(C4⋊C8), (C2×C4).84C42, (C2×C42).11C4, (C22×C8).14C4, C2.2(C16⋊C4), C8.49(C22⋊C4), C4.19(C22⋊C8), (C2×M5(2)).6C2, (C2×C4).68M4(2), C22.12(C8⋊C4), C22.22(C22⋊C8), (C22×C8).367C22, C4.25(C2.C42), C2.13(C22.7C42), (C2×C4).73(C2×C8), (C2×C8).237(C2×C4), (C2×C8⋊C4).17C2, (C2×C4).104(C4⋊C4), (C22×C4).466(C2×C4), (C2×C4).346(C22⋊C4), SmallGroup(128,107)

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

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

Generators and relations for C42.2C8
 G = < a,b,c | a4=b4=1, c8=b2, ab=ba, cac-1=ab-1, bc=cb >

Subgroups: 104 in 70 conjugacy classes, 44 normal (24 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C16, C42, C42, C2×C8, C2×C8, C2×C8, C22×C4, C22×C4, C8⋊C4, C2×C16, M5(2), C2×C42, C22×C8, C2×C8⋊C4, C2×M5(2), C42.2C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C2.C42, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C22.7C42, C16⋊C4, C42.2C8

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J8A···8H8I8J8K8L16A···16P
order12222244444444448···8888816···16
size11112211112244442···244444···4

44 irreducible representations

dim1111111122224
type++++-
imageC1C2C2C4C4C4C8C8D4Q8M4(2)M4(2)C16⋊C4
kernelC42.2C8C2×C8⋊C4C2×M5(2)C2×C16C2×C42C22×C8C42C2×C8C2×C8C2×C8C2×C4C23C2
# reps1128228831224

Matrix representation of C42.2C8 ►in GL6(𝔽17)

1300000
040000
001000
00141600
0000130
00100164
,
1600000
0160000
0013000
0001300
0000130
0000013
,
020000
900000
000010
009091
00141500
008480

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,4,0,0,0,0,0,0,1,14,0,10,0,0,0,16,0,0,0,0,0,0,13,16,0,0,0,0,0,4],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[0,9,0,0,0,0,2,0,0,0,0,0,0,0,0,9,14,8,0,0,0,0,15,4,0,0,1,9,0,8,0,0,0,1,0,0] >;
 

C42.2C8 in GAP, Magma, Sage, TeX 

C_4^2._2C_8
 
% in TeX
 
G:=Group("C4^2.2C8");
 
// GroupNames label
 
G:=SmallGroup(128,107);
 
// by ID
 
G=gap.SmallGroup(128,107);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,352,1018,136,2804,124]);
 
// Polycyclic
 
G:=Group<a,b,c|a^4=b^4=1,c^8=b^2,a*b=b*a,c*a*c^-1=a*b^-1,b*c=c*b>;
 
// generators/relations
 

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