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G = C42.4Q8  order 128 = 27

4th non-split extension by C42 of Q8 acting via Q8/C2=C22

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

Aliases: C42.4Q8, C42.21D4, C4⋊C8⋊1C4, C8⋊C4⋊3C4, C4.29C4≀C2, (C2×C42).1C4, C42.30(C2×C4), (C2×C4).29C42, (C22×C4).175D4, C2.5(C42⋊6C4), C2.4(C4.9C42), C42.6C4.4C2, (C2×C42).125C22, C22.4(C4.D4), C2.3(C22.C42), C23.137(C22⋊C4), C22.4(C4.10D4), C22.35(C2.C42), (C4×C4⋊C4).2C2, (C2×C4).69(C4⋊C4), (C22×C4).426(C2×C4), (C2×C4).293(C22⋊C4), SmallGroup(128,17)

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

Derived series
C1 — C2×C4 — C42.4Q8
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C4⋊C4 — C42.4Q8
Lower central
C1 — C2 — C2×C4 — C42.4Q8
Upper central
C1 — C22 — C2×C42 — C42.4Q8
Jennings
C1 — C22 — C22 — C2×C42 — C42.4Q8

Generators and relations for C42.4Q8
 G = < a,b,c,d | a4=b4=c4=1, d2=bc2, ab=ba, ac=ca, dad-1=a-1b2, cbc-1=a2b-1, dbd-1=a2b, dcd-1=a-1b2c-1 >

Subgroups: 160 in 82 conjugacy classes, 34 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C23, C42, C42, C4⋊C4, C2×C8, C22×C4, C22×C4, C2.C42, C8⋊C4, C8⋊C4, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C4×C4⋊C4, C42.6C4, C42.4Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2.C42, C4.D4, C4.10D4, C4≀C2, C4.9C42, C42⋊6C4, C22.C42, C42.4Q8

Smallest permutation representation of C42.4Q8
►On 32 points
Generators in S32
(1 15 27 18)(2 23 28 12)(3 9 29 20)(4 17 30 14)(5 11 31 22)(6 19 32 16)(7 13 25 24)(8 21 26 10)

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

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

(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)| (1,15,27,18)(2,23,28,12)(3,9,29,20)(4,17,30,14)(5,11,31,22)(6,19,32,16)(7,13,25,24)(8,21,26,10), (1,29,5,25)(2,4,6,8)(3,31,7,27)(9,22,13,18)(10,12,14,16)(11,24,15,20)(17,19,21,23)(26,28,30,32), (1,7,27,25)(2,10)(3,31,29,5)(4,19)(6,14)(8,23)(9,22,20,11)(12,26)(13,18,24,15)(16,30)(17,32)(21,28), (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( (1,15,27,18)(2,23,28,12)(3,9,29,20)(4,17,30,14)(5,11,31,22)(6,19,32,16)(7,13,25,24)(8,21,26,10), (1,29,5,25)(2,4,6,8)(3,31,7,27)(9,22,13,18)(10,12,14,16)(11,24,15,20)(17,19,21,23)(26,28,30,32), (1,7,27,25)(2,10)(3,31,29,5)(4,19)(6,14)(8,23)(9,22,20,11)(12,26)(13,18,24,15)(16,30)(17,32)(21,28), (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([[(1,15,27,18),(2,23,28,12),(3,9,29,20),(4,17,30,14),(5,11,31,22),(6,19,32,16),(7,13,25,24),(8,21,26,10)], [(1,29,5,25),(2,4,6,8),(3,31,7,27),(9,22,13,18),(10,12,14,16),(11,24,15,20),(17,19,21,23),(26,28,30,32)], [(1,7,27,25),(2,10),(3,31,29,5),(4,19),(6,14),(8,23),(9,22,20,11),(12,26),(13,18,24,15),(16,30),(17,32),(21,28)], [(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 2A2B2C2D2E4A···4H4I···4R8A···8H
order1222224···44···48···8
size1111222···24···48···8

32 irreducible representations

dim1111112222444
type++++-++-
imageC1C2C2C4C4C4D4Q8D4C4≀C2C4.D4C4.10D4C4.9C42
kernelC42.4Q8C4×C4⋊C4C42.6C4C8⋊C4C4⋊C8C2×C42C42C42C22×C4C4C22C22C2
# reps1124441128112

Matrix representation of C42.4Q8 ►in GL6(𝔽17)

400000
040000
0013000
0001300
000040
000004
,
400000
4130000
0016000
002100
000010
00001516
,
400000
1010000
001100
00151600
000044
0000913
,
1150000
11160000
000010
000001
001000
00151600

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,4,0,0,0,0,0,13,0,0,0,0,0,0,16,2,0,0,0,0,0,1,0,0,0,0,0,0,1,15,0,0,0,0,0,16],[4,10,0,0,0,0,0,1,0,0,0,0,0,0,1,15,0,0,0,0,1,16,0,0,0,0,0,0,4,9,0,0,0,0,4,13],[1,11,0,0,0,0,15,16,0,0,0,0,0,0,0,0,1,15,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

C42.4Q8 in GAP, Magma, Sage, TeX 

C_4^2._4Q_8
 
% in TeX
 
G:=Group("C4^2.4Q8");
 
// GroupNames label
 
G:=SmallGroup(128,17);
 
// by ID
 
G=gap.SmallGroup(128,17);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,520,1018,136,3924]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b*c^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^2*b,d*c*d^-1=a^-1*b^2*c^-1>;
 
// generators/relations
 

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