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G = C2×C24⋊C22  order 128 = 27

Direct product of C2 and C24⋊C22

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

Aliases: C2×C24⋊C22, C25⋊7C22, C24⋊9C23, C42⋊16C23, C23.58C24, C22.115C25, C22.1182+ (1+4), (C2×Q8)⋊9C23, C22⋊C4⋊11C23, (C2×C4).105C24, (C2×C42)⋊69C22, C22≀C2⋊37C22, (C2×D4).309C23, C4.4D4⋊88C22, (C22×Q8)⋊37C22, C2.46(C2×2+ (1+4)), (C22×C4).1213C23, (C22×D4).432C22, (C2×C22≀C2)⋊28C2, (C2×C4.4D4)⋊57C2, (C2×C22⋊C4)⋊55C22, SmallGroup(128,2258)

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

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

Subgroups: 1436 in 704 conjugacy classes, 388 normal (4 characteristic)
C1, C2 [×7], C2 [×12], C4 [×18], C22, C22 [×6], C22 [×92], C2×C4 [×18], C2×C4 [×18], D4 [×36], Q8 [×12], C23, C23 [×12], C23 [×92], C42 [×12], C22⋊C4 [×72], C22×C4 [×9], C2×D4 [×36], C2×D4 [×18], C2×Q8 [×12], C2×Q8 [×6], C24 [×14], C24 [×12], C2×C42 [×3], C2×C22⋊C4 [×18], C22≀C2 [×48], C4.4D4 [×72], C22×D4 [×9], C22×Q8 [×3], C25 [×2], C2×C22≀C2 [×6], C2×C4.4D4 [×9], C24⋊C22 [×16], C2×C24⋊C22

Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], C24 [×31], 2+ (1+4) [×6], C25, C24⋊C22 [×4], C2×2+ (1+4) [×3], C2×C24⋊C22

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, fbf=be=eb, gbg=bde, gcg=cd=dc, ce=ec, fcf=cde, de=ed, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >

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

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

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

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

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

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

Matrix representation ►G ⊆ GL12(ℤ)

100000000000
010000000000
001000000000
000100000000
0000-10000000
00000-1000000
000000-100000
0000000-10000
000000001000
000000000100
000000000010
000000000001
,
100000000000
1-10000000000
10-1000000000
000100000000
0000-10000000
0000-11000000
0000-10100000
0000000-10000
00000000-1000
000000000-100
000000000010
000000000001
,
-100000000000
0-10000000000
-101000000000
-100100000000
000010000000
000001000000
000010-100000
0000100-10000
000000001000
000000000-100
000000000010
00000000000-1
,
100000000000
010000000000
001000000000
000100000000
000010000000
000001000000
000000100000
000000010000
00000000-1000
000000000-100
0000000000-10
00000000000-1
,
-100000000000
0-10000000000
00-1000000000
000-100000000
0000-10000000
00000-1000000
000000-100000
0000000-10000
00000000-1000
000000000-100
0000000000-10
00000000000-1
,
-102000000000
001-100000000
001000000000
0-11000000000
000010-200000
000000-110000
000000-100000
000001-100000
000000000010
000000000001
000000001000
000000000100
,
-120000000000
010000000000
010-100000000
01-1000000000
00001-2000000
00000-1000000
00000-1010000
00000-1100000
000000000100
000000001000
000000000001
000000000010

G:=sub<GL(12,Integers())| [1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[1,1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[-1,0,-1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,2,1,1,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[-1,0,0,0,0,0,0,0,0,0,0,0,2,1,1,1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0] >;
 

38 conjugacy classes

class 1 2A···2G2H···2S4A···4R
order12···22···24···4
size11···14···44···4

38 irreducible representations

dim11114
type+++++
imageC1C2C2C22+ (1+4)
kernelC2×C24⋊C22C2×C22≀C2C2×C4.4D4C24⋊C22C22
# reps169166

In GAP, Magma, Sage, TeX 

C_2\times C_2^4\rtimes C_2^2
 
% in TeX
 
G:=Group("C2xC2^4:C2^2");
 
// GroupNames label
 
G:=SmallGroup(128,2258);
 
// by ID
 
G=gap.SmallGroup(128,2258);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,232,1430,1059,2915,570]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=g^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,f*b*f=b*e=e*b,g*b*g=b*d*e,g*c*g=c*d=d*c,c*e=e*c,f*c*f=c*d*e,d*e=e*d,d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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