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## G = C2×2+ 1+4order 64 = 26

### Direct product of C2 and 2+ 1+4

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

Aliases: C2×2+ 1+4, D44C23, C2.4C25, Q84C23, C4.12C24, C232C23, C245C22, C22.2C24, D4(C2×D4), Q8(C2×Q8), (C2×C4)⋊2C23, C4○D46C22, (C2×D4)⋊17C22, (C22×D4)⋊12C2, (C2×Q8)⋊20C22, (C22×C4)⋊13C22, (C2×D4)(C2×D4), (C2×Q8)(C2×Q8), (C2×C4○D4)⋊13C2, SmallGroup(64,264)

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2×2+ 1+4
G = < a,b,c,d,e | a2=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >

Subgroups: 593 in 449 conjugacy classes, 377 normal (4 characteristic)
C1, C2, C2 [×2], C2 [×18], C4 [×12], C22, C22 [×18], C22 [×42], C2×C4 [×42], D4 [×72], Q8 [×8], C23 [×33], C23 [×12], C22×C4 [×9], C2×D4 [×90], C2×Q8 [×2], C4○D4 [×48], C24 [×6], C22×D4 [×9], C2×C4○D4 [×6], 2+ 1+4 [×16], C2×2+ 1+4
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C24 [×31], 2+ 1+4 [×2], C25, C2×2+ 1+4

Permutation representations of C2×2+ 1+4
On 16 points - transitive group 16T69
Generators in S16
(1 11)(2 12)(3 9)(4 10)(5 14)(6 15)(7 16)(8 13)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 4)(2 3)(5 8)(6 7)(9 12)(10 11)(13 14)(15 16)
(1 16 3 14)(2 13 4 15)(5 11 7 9)(6 12 8 10)
(1 14)(2 15)(3 16)(4 13)(5 11)(6 12)(7 9)(8 10)

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

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

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

G:=TransitiveGroup(16,69);

34 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2U 4A ··· 4L order 1 2 2 2 2 ··· 2 4 ··· 4 size 1 1 1 1 2 ··· 2 2 ··· 2

34 irreducible representations

 dim 1 1 1 1 4 type + + + + + image C1 C2 C2 C2 2+ 1+4 kernel C2×2+ 1+4 C22×D4 C2×C4○D4 2+ 1+4 C2 # reps 1 9 6 16 2

Matrix representation of C2×2+ 1+4 in GL5(ℤ)

 -1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 -1 0 0 0 1 0 0 0 -1 0 0 0 1 0 0 0
,
 -1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 -1
,
 1 0 0 0 0 0 0 -1 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 1 0
,
 -1 0 0 0 0 0 0 -1 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,Integers())| [-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,-1,0,0,0,1,0,0,0,-1,0,0,0],[-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1],[1,0,0,0,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,-1,0],[-1,0,0,0,0,0,0,-1,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,1,0] >;

C2×2+ 1+4 in GAP, Magma, Sage, TeX

C_2\times 2_+^{1+4}
% in TeX

G:=Group("C2xES+(2,2)");
// GroupNames label

G:=SmallGroup(64,264);
// by ID

G=gap.SmallGroup(64,264);
# by ID

G:=PCGroup([6,-2,2,2,2,2,-2,409,332,963]);
// Polycyclic

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

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