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## G = C22×C22≀C2order 128 = 27

### Direct product of C22 and C22≀C2

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

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

Generators and relations for C22×C22≀C2
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 >

Subgroups: 4076 in 2272 conjugacy classes, 556 normal (6 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C22⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C24, C2×C22⋊C4, C22≀C2, C23×C4, C22×D4, C22×D4, C25, C25, C25, C22×C22⋊C4, C2×C22≀C2, D4×C23, C26, C22×C22≀C2
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22≀C2, C22×D4, C25, C2×C22≀C2, D4×C23, C22×C22≀C2

Smallest permutation representation of C22×C22≀C2
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 4)(2 3)(5 7)(6 8)(9 23)(10 24)(11 13)(12 14)(15 18)(16 17)(19 21)(20 22)(25 27)(26 28)(29 31)(30 32)
(1 5)(2 6)(3 8)(4 7)(9 14)(10 13)(11 24)(12 23)(15 28)(16 27)(17 25)(18 26)(19 31)(20 32)(21 29)(22 30)
(1 23)(2 24)(3 10)(4 9)(5 12)(6 11)(7 14)(8 13)(15 16)(17 18)(19 20)(21 22)(25 26)(27 28)(29 30)(31 32)
(1 13)(2 14)(3 12)(4 11)(5 10)(6 9)(7 24)(8 23)(15 31)(16 32)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)
(1 24)(2 23)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(15 26)(16 25)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)
(1 20)(2 19)(3 21)(4 22)(5 16)(6 15)(7 17)(8 18)(9 31)(10 32)(11 25)(12 26)(13 27)(14 28)(23 29)(24 30)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2O 2P ··· 2AM 2AN 2AO 2AP 2AQ 4A ··· 4L order 1 2 ··· 2 2 ··· 2 2 2 2 2 4 ··· 4 size 1 1 ··· 1 2 ··· 2 4 4 4 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 2 type + + + + + + image C1 C2 C2 C2 C2 D4 kernel C22×C22≀C2 C22×C22⋊C4 C2×C22≀C2 D4×C23 C26 C24 # reps 1 3 24 3 1 24

Matrix representation of C22×C22≀C2 in GL6(ℤ)

 -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 -1 -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 -2 0 0 0 0 0 1

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] >;

C22×C22≀C2 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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