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G = C2×C23⋊2Q8  order 128 = 27

Direct product of C2 and C23⋊2Q8

direct product, p-group, metabelian, nilpotent (class 2), monomial, rational

Aliases: C2×C23⋊2Q8, C24⋊7Q8, C25.75C22, C22.45C25, C24.613C23, C23.117C24, C22.1062+ (1+4), C4⋊C4⋊6C23, C23⋊4(C2×Q8), (C2×Q8)⋊6C23, C2.7(Q8×C23), (C2×C4).47C24, C22⋊Q8⋊79C22, C22.5(C22×Q8), C22⋊C4.75C23, (C22×Q8)⋊27C22, (C23×C4).589C22, C2.11(C2×2+ (1+4)), (C22×C4).1185C23, (C2×C4⋊C4)⋊65C22, (C2×C22⋊Q8)⋊66C2, (C22×C22⋊C4).29C2, (C2×C22⋊C4).534C22, SmallGroup(128,2188)

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

Derived series
C1 — C22 — C2×C23⋊2Q8
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C22⋊C4 — C2×C23⋊2Q8
Lower central
C1 — C22 — C2×C23⋊2Q8
Upper central
C1 — C23 — C2×C23⋊2Q8
Jennings
C1 — C22 — C2×C23⋊2Q8

Subgroups: 1068 in 648 conjugacy classes, 436 normal (6 characteristic)
C1, C2, C2 [×6], C2 [×12], C4 [×24], C22, C22 [×18], C22 [×52], C2×C4 [×24], C2×C4 [×48], Q8 [×16], C23, C23 [×34], C23 [×36], C22⋊C4 [×48], C4⋊C4 [×48], C22×C4 [×36], C22×C4 [×12], C2×Q8 [×16], C2×Q8 [×8], C24 [×15], C24 [×4], C2×C22⋊C4 [×36], C2×C4⋊C4 [×12], C22⋊Q8 [×96], C23×C4 [×6], C22×Q8 [×4], C25, C22×C22⋊C4 [×3], C2×C22⋊Q8 [×12], C23⋊2Q8 [×16], C2×C23⋊2Q8

Quotients:
C1, C2 [×31], C22 [×155], Q8 [×8], C23 [×155], C2×Q8 [×28], C24 [×31], C22×Q8 [×14], 2+ (1+4) [×4], C25, C23⋊2Q8 [×4], Q8×C23, C2×2+ (1+4) [×2], C2×C23⋊2Q8

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=fbf-1=bd=db, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e-1 >

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

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

(13 28)(14 25)(15 26)(16 27)(17 31)(18 32)(19 29)(20 30)

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

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

Matrix representation ►G ⊆ GL8(𝔽5)

10000000
01000000
00400000
00040000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00004000
00000100
00000010
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000040
00000004
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
04000000
10000000
00010000
00400000
00000400
00001000
00000001
00000040
,
20000000
03000000
00200000
00030000
00000040
00000004
00001000
00000100

G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0],[2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0] >;
 

44 conjugacy classes

class 1 2A···2G2H···2S4A···4X
order12···22···24···4
size11···12···24···4

44 irreducible representations

dim111124
type++++-+
imageC1C2C2C2Q82+ (1+4)
kernelC2×C23⋊2Q8C22×C22⋊C4C2×C22⋊Q8C23⋊2Q8C24C22
# reps13121684

In GAP, Magma, Sage, TeX 

C_2\times C_2^3\rtimes_2Q_8
 
% in TeX
 
G:=Group("C2xC2^3:2Q8");
 
// GroupNames label
 
G:=SmallGroup(128,2188);
 
// by ID
 
G=gap.SmallGroup(128,2188);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,1430,387,352,1123]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=1,f^2=e^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,e*b*e^-1=f*b*f^-1=b*d=d*b,f*c*f^-1=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
 
// generators/relations
 

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