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G = C42.96D4  order 128 = 27

78th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.96D4, (C2×D4).21Q8, (C2×D4).197D4, C4⋊2(C4.D4), C23.8(C4⋊C4), (C23×C4).21C4, C24.30(C2×C4), C4.91(C22⋊Q8), C4.116(C4⋊D4), C4⋊M4(2)⋊23C2, C22.C42⋊12C2, C23.186(C22×C4), (C2×C42).245C22, (C22×C4).660C23, C2.9(C23.7Q8), (C22×D4).455C22, C22.26(C42⋊C2), (C2×M4(2)).151C22, C2.23(M4(2).8C22), (C2×C4×D4).14C2, (C2×C4).1(C2×Q8), C22.18(C2×C4⋊C4), (C2×C4).45(C4○D4), (C2×C4).1312(C2×D4), (C2×C22⋊C4).37C4, (C22×C4).50(C2×C4), (C2×C4.D4).6C2, C2.25(C2×C4.D4), (C2×C4⋊C4).753C22, (C2×C4).356(C22⋊C4), C22.245(C2×C22⋊C4), SmallGroup(128,532)

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

Derived series
C1 — C23 — C42.96D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C2×C4×D4 — C42.96D4
Lower central
C1 — C2 — C23 — C42.96D4
Upper central
C1 — C22 — C2×C42 — C42.96D4
Jennings
C1 — C2 — C2 — C22×C4 — C42.96D4

Generators and relations for C42.96D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, bd=db, dcd-1=b-1c3 >

Subgroups: 356 in 168 conjugacy classes, 62 normal (20 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C4.D4, C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C2×M4(2), C23×C4, C22×D4, C22.C42, C2×C4.D4, C4⋊M4(2), C2×C4×D4, C42.96D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C4.D4, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, C23.7Q8, C2×C4.D4, M4(2).8C22, C42.96D4

Smallest permutation representation of C42.96D4
►On 32 points
Generators in S32
(1 16 25 19)(2 24 26 13)(3 10 27 21)(4 18 28 15)(5 12 29 23)(6 20 30 9)(7 14 31 17)(8 22 32 11)

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

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

32 conjugacy classes

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

32 irreducible representations

dim1111111222244
type+++++++-+
imageC1C2C2C2C2C4C4D4D4Q8C4○D4C4.D4M4(2).8C22
kernelC42.96D4C22.C42C2×C4.D4C4⋊M4(2)C2×C4×D4C2×C22⋊C4C23×C4C42C2×D4C2×D4C2×C4C4C2
# reps1222144422422

Matrix representation of C42.96D4 ►in GL6(𝔽17)

0160000
100000
0016000
0001600
000010
000001
,
1600000
0160000
000100
0016000
000001
0000160
,
1300000
040000
000001
000010
001000
0001600
,
1300000
040000
000010
000001
000100
0016000

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

C42.96D4 in GAP, Magma, Sage, TeX 

C_4^2._{96}D_4
 
% in TeX
 
G:=Group("C4^2.96D4");
 
// GroupNames label
 
G:=SmallGroup(128,532);
 
// by ID
 
G=gap.SmallGroup(128,532);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,2804,1027,124]);
 
// 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^-1*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^3>;
 
// generators/relations
 

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