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G = C2×C2≀C22order 128 = 27

Direct product of C2 and C2≀C22

direct product, p-group, metabelian, nilpotent (class 3), monomial, rational

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C2×C2≀C22
 Chief series C1 — C2 — C22 — C23 — C24 — C22×D4 — C2×2+ 1+4 — C2×C2≀C22
 Lower central C1 — C2 — C23 — C2×C2≀C22
 Upper central C1 — C22 — C24 — C2×C2≀C22
 Jennings C1 — C2 — C23 — C2×C2≀C22

Generators and relations for C2×C2≀C22
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=fbf=bcd, ece-1=cd=dc, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 1236 in 509 conjugacy classes, 108 normal (8 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C22⋊C4, C22⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C24, C24, C23⋊C4, C2×C22⋊C4, C2×C22⋊C4, C22≀C2, C22≀C2, C22×D4, C22×D4, C2×C4○D4, 2+ 1+4, 2+ 1+4, C25, C2×C23⋊C4, C2≀C22, C2×C22≀C2, C2×2+ 1+4, C2×C2≀C22
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22≀C2, C22×D4, C2≀C22, C2×C22≀C2, C2×C2≀C22

Permutation representations of C2×C2≀C22
On 16 points - transitive group 16T245
Generators in S16
(1 6)(2 5)(3 7)(4 8)(9 14)(10 15)(11 16)(12 13)
(1 11)(2 13)(3 14)(4 10)(5 12)(6 16)(7 9)(8 15)
(1 3)(2 5)(4 8)(6 7)(9 16)(10 15)(11 14)(12 13)
(1 7)(2 8)(3 6)(4 5)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 3)(2 4)(5 8)(6 7)(9 11)(14 16)

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

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

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

G:=TransitiveGroup(16,245);

On 16 points - transitive group 16T246
Generators in S16
(1 5)(2 6)(3 7)(4 8)(9 16)(10 13)(11 14)(12 15)
(1 11)(2 10)(3 15)(4 16)(5 14)(6 13)(7 12)(8 9)
(2 7)(3 6)(10 12)(13 15)
(1 8)(2 7)(3 6)(4 5)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(9 11)(14 16)

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

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

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

G:=TransitiveGroup(16,246);

On 16 points - transitive group 16T271
Generators in S16
(1 4)(2 3)(5 8)(6 7)(9 15)(10 16)(11 13)(12 14)
(1 16)(2 14)(3 12)(4 10)(5 13)(6 15)(7 9)(8 11)
(1 5)(2 6)(3 7)(4 8)(9 12)(10 11)(13 16)(14 15)
(1 2)(3 4)(5 6)(7 8)(9 11)(10 12)(13 15)(14 16)
(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 4)(2 3)(5 8)(6 7)(9 16)(10 15)(11 14)(12 13)

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

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

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

G:=TransitiveGroup(16,271);

32 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J ··· 2S 4A ··· 4F 4G ··· 4L order 1 2 2 2 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 1 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8

32 irreducible representations

 dim 1 1 1 1 1 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 D4 D4 D4 C2≀C22 kernel C2×C2≀C22 C2×C23⋊C4 C2≀C22 C2×C22≀C2 C2×2+ 1+4 C22×C4 C2×D4 C24 C2 # reps 1 3 8 3 1 3 6 3 4

Matrix representation of C2×C2≀C22 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
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 2 0 0 1 0 0 0 0 0 -1 0 -1 -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 1 1 1 2 0 0 -1 -1 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 1 1 2 0 0 0 0 -1 0 0 0 -1 0 0 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 1 1 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],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,-1,0,0,0,1,0,0,0,0,1,1,0,-1,0,0,0,2,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,1,-1,0,0,1,0,1,-1,0,0,0,0,1,0,0,0,0,0,2,-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,-1,0,0,0,1,0,0,0,0,0,1,-1,0,0,0,0,2,0,0,-1],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,-1,1,0,0,0,0,0,1,0,0,0,0,0,2,-1] >;

C2×C2≀C22 in GAP, Magma, Sage, TeX

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

G:=Group("C2xC2wrC2^2");
// GroupNames label

G:=SmallGroup(128,1755);
// by ID

G=gap.SmallGroup(128,1755);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,718,2028]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=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,b*d=d*b,e*b*e^-1=f*b*f=b*c*d,e*c*e^-1=c*d=d*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations

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