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G = C2×C23⋊3D4  order 128 = 27

Direct product of C2 and C23⋊3D4

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

Aliases: C2×C23⋊3D4, C24⋊12D4, C24⋊7C23, C25⋊4C22, C22.34C25, C23.115C24, C22.992+ (1+4), C4⋊C4⋊4C23, C23⋊6(C2×D4), (C2×D4)⋊5C23, (D4×C23)⋊12C2, C22⋊C4⋊4C23, (C2×C4).37C24, C2.13(D4×C23), C4⋊D4⋊60C22, (C22×C4)⋊13C23, (C23×C4)⋊29C22, C22≀C2⋊25C22, (C22×D4)⋊29C22, C22.47(C22×D4), C2.4(C2×2+ (1+4)), C22.D4⋊30C22, (C2×C4⋊D4)⋊53C2, (C2×C4⋊C4)⋊63C22, (C2×C22≀C2)⋊20C2, (C22×C22⋊C4)⋊30C2, (C2×C22⋊C4)⋊39C22, (C2×C22.D4)⋊48C2, SmallGroup(128,2177)

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

Derived series
C1 — C22 — C2×C23⋊3D4
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — D4×C23 — C2×C23⋊3D4
Lower central
C1 — C22 — C2×C23⋊3D4
Upper central
C1 — C23 — C2×C23⋊3D4
Jennings
C1 — C22 — C2×C23⋊3D4

Subgroups: 1932 in 992 conjugacy classes, 436 normal (8 characteristic)
C1, C2, C2 [×6], C2 [×20], C4 [×16], C22, C22 [×18], C22 [×124], C2×C4 [×16], C2×C4 [×40], D4 [×80], C23, C23 [×42], C23 [×124], C22⋊C4 [×48], C4⋊C4 [×16], C22×C4 [×24], C22×C4 [×12], C2×D4 [×48], C2×D4 [×104], C24, C24 [×26], C24 [×20], C2×C22⋊C4 [×20], C2×C4⋊C4 [×4], C22≀C2 [×32], C4⋊D4 [×32], C22.D4 [×32], C23×C4 [×4], C22×D4 [×28], C22×D4 [×16], C25, C25 [×2], C22×C22⋊C4, C2×C22≀C2 [×4], C2×C4⋊D4 [×4], C2×C22.D4 [×4], C23⋊3D4 [×16], D4×C23 [×2], C2×C23⋊3D4

Quotients:
C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C24 [×31], C22×D4 [×14], 2+ (1+4) [×4], C25, C23⋊3D4 [×4], D4×C23, C2×2+ (1+4) [×2], C2×C23⋊3D4

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

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

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

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

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

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

Matrix representation ►G ⊆ GL8(ℤ)

-10000000
0-1000000
00100000
00010000
0000-1000
00000-100
000000-10
0000000-1
,
10000000
01000000
00-100000
000-10000
00000-100
0000-1000
0000-1001
00000-110
,
10000000
01000000
00100000
00010000
0000-1000
00000-100
00001010
0000-2-101
,
10000000
01000000
00100000
00010000
0000-1000
00000-100
000000-10
0000000-1
,
0-1000000
10000000
000-10000
00100000
00001020
0000210-2
0000-10-10
0000212-1
,
01000000
10000000
00010000
00100000
00001020
0000-2-102
000000-10
00000021

G:=sub<GL(8,Integers())| [-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,-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],[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,-1,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,-1,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,1,-2,0,0,0,0,0,-1,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[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,-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],[0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,2,-1,2,0,0,0,0,0,1,0,1,0,0,0,0,2,0,-1,2,0,0,0,0,0,-2,0,-1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,-2,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,2,0,-1,2,0,0,0,0,0,2,0,1] >;
 

44 conjugacy classes

class 1 2A···2G2H···2S2T···2AA4A···4P
order12···22···22···24···4
size11···12···24···44···4

44 irreducible representations

dim111111124
type+++++++++
imageC1C2C2C2C2C2C2D42+ (1+4)
kernelC2×C23⋊3D4C22×C22⋊C4C2×C22≀C2C2×C4⋊D4C2×C22.D4C23⋊3D4D4×C23C24C22
# reps1144416284

In GAP, Magma, Sage, TeX 

C_2\times C_2^3\rtimes_3D_4
 
% in TeX
 
G:=Group("C2xC2^3:3D4");
 
// GroupNames label
 
G:=SmallGroup(128,2177);
 
// by ID
 
G=gap.SmallGroup(128,2177);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,387,1123]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f=b*d=d*b,b*e=e*b,e*c*e^-1=f*c*f=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
 
// generators/relations
 

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