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G = C42⋊6C8  order 128 = 27

3rd semidirect product of C42 and C8 acting via C8/C4=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C42⋊6C8, C43.5C2, C42.41Q8, C42.453D4, C4⋊C8⋊5C4, C4.1(C4×C8), C4.36C4≀C2, C4.22(C4⋊C8), C4.1(C8⋊C4), (C2×C42).41C4, (C2×C4).49C42, C42.250(C2×C4), (C22×C4).634D4, (C2×C4).65M4(2), C2.1(C42⋊6C4), C22.9(C22⋊C8), C42.12C4.3C2, C23.134(C22⋊C4), (C2×C42).1024C22, C2.5(C22.7C42), C22.17(C2.C42), (C2×C4).68(C2×C8), (C2×C4).155(C4⋊C4), (C22×C4).462(C2×C4), (C2×C4).368(C22⋊C4), SmallGroup(128,8)

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

Derived series
C1 — C4 — C42⋊6C8
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C43 — C42⋊6C8
Lower central
C1 — C2 — C4 — C42⋊6C8
Upper central
C1 — C42 — C2×C42 — C42⋊6C8
Jennings
C1 — C22 — C22 — C2×C42 — C42⋊6C8

Generators and relations for C42⋊6C8
 G = < a,b,c | a4=b4=c8=1, cac-1=ab=ba, cbc-1=b-1 >

Subgroups: 168 in 104 conjugacy classes, 48 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C42, C2×C8, C22×C4, C22×C4, C22×C4, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C2×C42, C2×C42, C43, C42.12C4, C42⋊6C8
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≀C2, C4⋊C8, C22.7C42, C42⋊6C4, C42⋊6C8

Smallest permutation representation of C42⋊6C8
►On 32 points
Generators in S32
(2 22 32 12)(4 24 26 14)(6 18 28 16)(8 20 30 10)

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A···4L4M···4AH8A···8P
order1222224···44···48···8
size1111221···12···24···4

56 irreducible representations

dim11111122222
type++++-+
imageC1C2C2C4C4C8D4Q8D4M4(2)C4≀C2
kernelC42⋊6C8C43C42.12C4C4⋊C8C2×C42C42C42C42C22×C4C2×C4C4
# reps1128416112416

Matrix representation of C42⋊6C8 ►in GL4(𝔽17) generated by

1400
01600
0010
0004
,
16000
01600
0040
00013
,
9000
4800
0001
00160
G:=sub<GL(4,GF(17))| [1,0,0,0,4,16,0,0,0,0,1,0,0,0,0,4],[16,0,0,0,0,16,0,0,0,0,4,0,0,0,0,13],[9,4,0,0,0,8,0,0,0,0,0,16,0,0,1,0] >;
 

C42⋊6C8 in GAP, Magma, Sage, TeX 

C_4^2\rtimes_6C_8
 
% in TeX
 
G:=Group("C4^2:6C8");
 
// GroupNames label
 
G:=SmallGroup(128,8);
 
// by ID
 
G=gap.SmallGroup(128,8);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,184,248,1684]);
 
// Polycyclic
 
G:=Group<a,b,c|a^4=b^4=c^8=1,c*a*c^-1=a*b=b*a,c*b*c^-1=b^-1>;
 
// generators/relations
 

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