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## G = C2×C22≀C2order 64 = 26

### Direct product of C2 and C22≀C2

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

Aliases: C2×C22≀C2, C251C2, C235D4, C231C23, C246C22, C22.15C24, (C2×C4)⋊1C23, C223(C2×D4), (C2×D4)⋊9C22, (C22×D4)⋊3C2, C2.4(C22×D4), (C22×C4)⋊5C22, C22⋊C412C22, (C2×C22⋊C4)⋊8C2, SmallGroup(64,202)

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2×C22≀C2
G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf=bd=db, be=eb, cd=dc, fcf=ce=ec, de=ed, df=fd, ef=fe >

Subgroups: 569 in 331 conjugacy classes, 105 normal (6 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C22⋊C4, C22×C4, C2×D4, C2×D4, C24, C24, C24, C2×C22⋊C4, C22≀C2, C22×D4, C25, C2×C22≀C2
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22≀C2, C22×D4, C2×C22≀C2

Character table of C2×C22≀C2

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 2P 2Q 2R 2S 2T 2U 4A 4B 4C 4D 4E 4F size 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 -1 1 -1 1 1 1 linear of order 2 ρ3 1 -1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 -1 1 1 -1 -1 1 linear of order 2 ρ4 1 -1 1 1 1 -1 -1 -1 1 1 1 -1 -1 1 -1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 -1 1 linear of order 2 ρ5 1 -1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1 1 1 -1 -1 1 1 -1 linear of order 2 ρ6 1 1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 1 -1 -1 -1 linear of order 2 ρ7 1 -1 1 1 1 -1 -1 -1 1 1 1 -1 -1 1 -1 1 1 -1 -1 -1 1 -1 -1 -1 1 1 1 -1 linear of order 2 ρ8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ9 1 1 1 1 1 1 1 1 1 -1 -1 -1 1 1 -1 -1 -1 -1 1 -1 1 1 -1 1 -1 1 -1 -1 linear of order 2 ρ10 1 -1 1 1 1 -1 -1 -1 1 -1 -1 1 -1 1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 linear of order 2 ρ11 1 1 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 1 1 -1 -1 -1 1 -1 -1 1 1 1 1 -1 -1 linear of order 2 ρ12 1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 1 -1 -1 1 -1 1 1 -1 1 -1 1 1 -1 -1 1 -1 linear of order 2 ρ13 1 -1 1 1 1 -1 -1 -1 1 -1 -1 1 -1 1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 linear of order 2 ρ14 1 1 1 1 1 1 1 1 1 -1 -1 -1 1 1 -1 -1 -1 -1 1 -1 -1 -1 1 -1 1 -1 1 1 linear of order 2 ρ15 1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 1 -1 -1 1 -1 1 1 -1 -1 1 -1 -1 1 1 -1 1 linear of order 2 ρ16 1 1 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ17 2 -2 -2 -2 2 2 2 -2 -2 0 0 0 -2 2 0 0 0 0 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 -2 2 -2 -2 2 -2 2 0 -2 0 0 0 0 2 2 0 0 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 -2 -2 2 -2 -2 2 2 0 0 0 -2 -2 0 0 0 0 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 -2 2 -2 -2 2 -2 2 0 2 0 0 0 0 -2 -2 0 0 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 -2 -2 2 -2 -2 2 2 0 0 2 2 0 0 0 0 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ22 2 2 -2 -2 2 -2 -2 2 -2 0 0 0 2 2 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ23 2 2 2 -2 -2 -2 2 -2 0 -2 0 0 0 0 -2 2 0 0 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ24 2 2 -2 2 -2 2 -2 -2 0 0 -2 2 0 0 0 0 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ25 2 -2 -2 -2 2 2 2 -2 2 0 0 0 2 -2 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ26 2 2 -2 2 -2 2 -2 -2 0 0 2 -2 0 0 0 0 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ27 2 2 2 -2 -2 -2 2 -2 0 2 0 0 0 0 2 -2 0 0 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ28 2 -2 -2 2 -2 -2 2 2 0 0 -2 -2 0 0 0 0 2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4

Permutation representations of C2×C22≀C2
On 16 points - transitive group 16T105
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)
(1 3)(2 4)(5 12)(6 11)(7 14)(8 13)(9 15)(10 16)
(1 5)(2 6)(3 12)(4 11)(7 9)(8 10)(13 16)(14 15)
(1 5)(2 6)(3 12)(4 11)(7 15)(8 16)(9 14)(10 13)
(1 12)(2 11)(3 5)(4 6)(7 14)(8 13)(9 15)(10 16)
(1 10)(2 9)(3 8)(4 7)(5 13)(6 14)(11 15)(12 16)

G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,3)(2,4)(5,12)(6,11)(7,14)(8,13)(9,15)(10,16), (1,5)(2,6)(3,12)(4,11)(7,9)(8,10)(13,16)(14,15), (1,5)(2,6)(3,12)(4,11)(7,15)(8,16)(9,14)(10,13), (1,12)(2,11)(3,5)(4,6)(7,14)(8,13)(9,15)(10,16), (1,10)(2,9)(3,8)(4,7)(5,13)(6,14)(11,15)(12,16)>;

G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,3)(2,4)(5,12)(6,11)(7,14)(8,13)(9,15)(10,16), (1,5)(2,6)(3,12)(4,11)(7,9)(8,10)(13,16)(14,15), (1,5)(2,6)(3,12)(4,11)(7,15)(8,16)(9,14)(10,13), (1,12)(2,11)(3,5)(4,6)(7,14)(8,13)(9,15)(10,16), (1,10)(2,9)(3,8)(4,7)(5,13)(6,14)(11,15)(12,16) );

G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)], [(1,3),(2,4),(5,12),(6,11),(7,14),(8,13),(9,15),(10,16)], [(1,5),(2,6),(3,12),(4,11),(7,9),(8,10),(13,16),(14,15)], [(1,5),(2,6),(3,12),(4,11),(7,15),(8,16),(9,14),(10,13)], [(1,12),(2,11),(3,5),(4,6),(7,14),(8,13),(9,15),(10,16)], [(1,10),(2,9),(3,8),(4,7),(5,13),(6,14),(11,15),(12,16)]])

G:=TransitiveGroup(16,105);

Matrix representation of C2×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 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
,
 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

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,-1,0,0,0,0,0,0,1],[-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],[0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C2×C22≀C2 in GAP, Magma, Sage, TeX

C_2\times C_2^2\wr C_2
% in TeX

G:=Group("C2xC2^2wrC2");
// GroupNames label

G:=SmallGroup(64,202);
// by ID

G=gap.SmallGroup(64,202);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,217,650]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^2=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,c*d=d*c,f*c*f=c*e=e*c,d*e=e*d,d*f=f*d,e*f=f*e>;
// generators/relations

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