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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 [×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

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