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G = C22×C22≀C2  order 128 = 27

Direct product of C22 and C22≀C2

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

Aliases: C22×C22≀C2, C26⋊2C2, C24⋊16D4, C23⋊1C24, C25⋊10C22, C24⋊11C23, C22.20C25, (C2×C4)⋊1C24, (D4×C23)⋊9C2, C23⋊10(C2×D4), C2.4(D4×C23), (C2×D4)⋊13C23, C22⋊3(C22×D4), C22⋊C4⋊14C23, (C23×C4)⋊27C22, (C22×C4)⋊11C23, (C22×D4)⋊56C22, (C2×C22⋊C4)⋊80C22, (C22×C22⋊C4)⋊26C2, SmallGroup(128,2163)

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

Derived series
C1 — C22 — C22×C22≀C2
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — C26 — C22×C22≀C2
Lower central
C1 — C22 — C22×C22≀C2
Upper central
C1 — C24 — C22×C22≀C2
Jennings
C1 — C22 — C22×C22≀C2

Subgroups: 4076 in 2272 conjugacy classes, 556 normal (6 characteristic)
C1, C2 [×15], C2 [×28], C4 [×12], C22, C22 [×58], C22 [×332], C2×C4 [×12], C2×C4 [×36], D4 [×96], C23 [×103], C23 [×700], C22⋊C4 [×48], C22×C4 [×18], C22×C4 [×12], C2×D4 [×48], C2×D4 [×144], C24, C24 [×64], C24 [×308], C2×C22⋊C4 [×36], C22≀C2 [×64], C23×C4 [×3], C22×D4 [×36], C22×D4 [×24], C25, C25 [×15], C25 [×24], C22×C22⋊C4 [×3], C2×C22≀C2 [×24], D4×C23 [×3], C26, C22×C22≀C2

Quotients:
C1, C2 [×31], C22 [×155], D4 [×24], C23 [×155], C2×D4 [×84], C24 [×31], C22≀C2 [×16], C22×D4 [×42], C25, C2×C22≀C2 [×12], D4×C23 [×3], C22×C22≀C2

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, be=eb, bf=fb, bg=gb, cd=dc, gcg=ce=ec, cf=fc, de=ed, gdg=df=fd, 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 10)(2 9)(3 27)(4 28)(5 20)(6 19)(7 12)(8 11)(13 26)(14 25)(15 32)(16 31)(17 30)(18 29)(21 23)(22 24)

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

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

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

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

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

Matrix representation ►G ⊆ GL6(ℤ)

-100000
010000
00-1000
000-100
000010
000001
,
-100000
0-10000
001000
000100
0000-10
00000-1
,
100000
0-10000
001000
000-100
000010
0000-1-1
,
-100000
0-10000
001000
000-100
000010
000001
,
100000
010000
00-1000
000-100
0000-10
00000-1
,
100000
010000
00-1000
000-100
000010
000001
,
100000
0-10000
000-100
00-1000
0000-1-2
000001

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,-2,1] >;
 

56 conjugacy classes

class 1 2A···2O2P···2AM2AN2AO2AP2AQ4A···4L
order12···22···222224···4
size11···12···244444···4

56 irreducible representations

dim111112
type++++++
imageC1C2C2C2C2D4
kernelC22×C22≀C2C22×C22⋊C4C2×C22≀C2D4×C23C26C24
# reps13243124

In GAP, Magma, Sage, TeX 

C_2^2\times C_2^2\wr C_2
 
% in TeX
 
G:=Group("C2^2xC2^2wrC2");
 
// GroupNames label
 
G:=SmallGroup(128,2163);
 
// by ID
 
G=gap.SmallGroup(128,2163);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430]);
 
// 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,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,g*c*g=c*e=e*c,c*f=f*c,d*e=e*d,g*d*g=d*f=f*d,e*f=f*e,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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