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G = C42⋊Q8  order 128 = 27

1st semidirect product of C42 and Q8 acting via Q8/C2=C22

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

Aliases: C42⋊1Q8, C42.126D4, C4⋊Q8⋊28C4, C4.17(C4×Q8), C4.55(C4⋊Q8), C42.C2⋊12C4, C42⋊4C4.9C2, C42.168(C2×C4), (C22×C4).305D4, C23.575(C2×D4), C42⋊6C4.12C2, C4.95(C4.4D4), C4⋊M4(2).34C2, (C2×C42).338C22, C22.29(C22⋊Q8), (C22×C4).1429C23, C42⋊C2.45C22, C2.46(C42⋊C22), (C2×M4(2)).217C22, C23.37C23.21C2, C2.17(C23.67C23), C4⋊C4.102(C2×C4), (C2×C4).216(C2×Q8), (C2×C4).1550(C2×D4), (C2×C4).612(C4○D4), (C2×C4).443(C22×C4), (C2×C4).143(C22⋊C4), C22.304(C2×C22⋊C4), SmallGroup(128,727)

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

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

Generators and relations for C42⋊Q8
 G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, cac-1=ab2, dad-1=a-1b-1, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 228 in 122 conjugacy classes, 56 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×Q8, C2.C42, C4⋊C8, C2×C42, C2×C42, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C2×M4(2), C42⋊6C4, C42⋊4C4, C4⋊M4(2), C23.37C23, C42⋊Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C22⋊C4, C4×Q8, C22⋊Q8, C4.4D4, C4⋊Q8, C23.67C23, C42⋊C22, C42⋊Q8

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G···4R4S4T4U4V8A8B8C8D
order1222224444444···444448888
size1111221111224···488888888

32 irreducible representations

dim111111122224
type++++++-+
imageC1C2C2C2C2C4C4D4Q8D4C4○D4C42⋊C22
kernelC42⋊Q8C42⋊6C4C42⋊4C4C4⋊M4(2)C23.37C23C42.C2C4⋊Q8C42C42C22×C4C2×C4C2
# reps141114424244

Matrix representation of C42⋊Q8 ►in GL6(𝔽17)

1390000
440000
000100
0016000
000004
0000130
,
1600000
0160000
004000
000400
000040
000004
,
120000
16160000
0001600
0016000
000001
000010
,
10100000
1270000
000001
0000160
0001600
001000

G:=sub<GL(6,GF(17))| [13,4,0,0,0,0,9,4,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,13,0,0,0,0,4,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,16,0,0,0,0,2,16,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[10,12,0,0,0,0,10,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1,0,0,0] >;
 

C42⋊Q8 in GAP, Magma, Sage, TeX 

C_4^2\rtimes Q_8
 
% in TeX
 
G:=Group("C4^2:Q8");
 
// GroupNames label
 
G:=SmallGroup(128,727);
 
// by ID
 
G=gap.SmallGroup(128,727);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,400,422,723,100,2019,248,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=c^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

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