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G = C42.43D4  order 128 = 27

25th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.43D4, C23.3M4(2), C23⋊C8⋊14C2, C24.24(C2×C4), (C23×C4).18C4, (C22×D4).21C4, C42.6C4⋊22C2, C4.15(C4.D4), C22⋊C8.122C22, C23.167(C22×C4), (C2×C42).149C22, (C22×C4).430C23, C22.19(C2×M4(2)), C2.10(C24.4C4), C2.7(C23.C23), (C2×C4×D4).4C2, (C2×C4⋊C4).34C4, C2.6(C2×C4.D4), (C2×C4).1127(C2×D4), (C22×C4).45(C2×C4), (C2×C4).313(C22⋊C4), C22.148(C2×C22⋊C4), (C2×C22⋊C4).405C22, SmallGroup(128,198)

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

Derived series
C1 — C23 — C42.43D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C2×C4×D4 — C42.43D4
Lower central
C1 — C2 — C23 — C42.43D4
Upper central
C1 — C22 — C2×C42 — C42.43D4
Jennings
C1 — C2 — C22 — C22×C4 — C42.43D4

Generators and relations for C42.43D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b-1, bd=db, dcd-1=b-1c3 >

Subgroups: 332 in 150 conjugacy classes, 50 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C2×D4, C24, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C22×D4, C23⋊C8, C42.6C4, C2×C4×D4, C42.43D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, M4(2), C22×C4, C2×D4, C4.D4, C2×C22⋊C4, C2×M4(2), C24.4C4, C23.C23, C2×C4.D4, C42.43D4

Smallest permutation representation of C42.43D4
►On 32 points
Generators in S32
(1 15 29 17)(2 12 30 22)(3 9 31 19)(4 14 32 24)(5 11 25 21)(6 16 26 18)(7 13 27 23)(8 10 28 20)

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

(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)

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A···4H4I···4N8A···8H
order12222222224···44···48···8
size11112244442···24···48···8

32 irreducible representations

dim11111112244
type++++++
imageC1C2C2C2C4C4C4D4M4(2)C4.D4C23.C23
kernelC42.43D4C23⋊C8C42.6C4C2×C4×D4C2×C4⋊C4C23×C4C22×D4C42C23C4C2
# reps14212424822

Matrix representation of C42.43D4 ►in GL6(𝔽17)

0130000
1300000
0013000
0001300
0000130
0000013
,
1300000
0130000
001000
00161600
000010
00001616
,
7110000
6100000
000048
00001313
0013900
000400
,
7110000
6100000
000010
000001
001000
00161600

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

C42.43D4 in GAP, Magma, Sage, TeX 

C_4^2._{43}D_4
 
% in TeX
 
G:=Group("C4^2.43D4");
 
// GroupNames label
 
G:=SmallGroup(128,198);
 
// by ID
 
G=gap.SmallGroup(128,198);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,520,1123,851,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=b^-1*c^3>;
 
// generators/relations
 

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