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G = C2×2+ 1+4  order 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, D4⋊4C23, C2.4C25, Q8⋊4C23, C4.12C24, C23⋊2C23, C24⋊5C22, C22.2C24, D4○(C2×D4), Q8○(C2×Q8), (C2×C4)⋊2C23, C4○D4⋊6C22, (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, C2, C4, C22, C22, C22, C2×C4, D4, Q8, C23, C23, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, C2×2+ 1+4
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, 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);
 

C2×2+ 1+4 is a maximal subgroup of
 2+ 1+4⋊2C4  2+ 1+4⋊3C4  C4○D4.D4  C23.C24  M4(2).24C23  2+ 1+4⋊5C4  C4○D4⋊D4  D4.(C2×D4)  (C2×D4)⋊21D4  M4(2)⋊C23  C24⋊C23  C22.73C25  C22.74C25  C22.77C25  C4⋊2+ 1+4  C22.87C25  C22.89C25  D8⋊C23  2+ 1+6
C2×2+ 1+4 is a maximal quotient of
 C22.48C25  C22.49C25  C2×D42  C2×Q82  C22.70C25  C22.72C25  C22.73C25  C22.77C25  C22.79C25  C22.81C25  C22.83C25  C4⋊2+ 1+4  C22.87C25  C22.90C25  C22.92C25  C22.94C25  C22.95C25  C22.97C25  C22.100C25  C22.102C25  C22.103C25  C22.106C25  C22.108C25  C23.144C24  C22.111C25  C22.118C25  C42⋊C23  C22.122C25  C22.123C25  C22.124C25  C22.125C25  C22.126C25  C22.127C25  C22.128C25  C22.129C25  C22.130C25  C22.131C25  C22.132C25  C22.133C25  C22.134C25  C22.135C25  C22.136C25  C22.137C25  C22.138C25  C22.139C25  C22.140C25  C22.141C25  C22.143C25  C22.147C25  C22.148C25  C22.149C25  C22.150C25  C22.151C25

34 conjugacy classes

class 1 2A2B2C2D···2U4A···4L
order12222···24···4
size11112···22···2

34 irreducible representations

dim11114
type+++++
imageC1C2C2C22+ 1+4
kernelC2×2+ 1+4C22×D4C2×C4○D42+ 1+4C2
# reps196162

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

-10000
01000
00100
00010
00001
,
10000
0000-1
00010
00-100
01000
,
-10000
01000
00100
000-10
0000-1
,
10000
00-100
01000
0000-1
00010
,
-10000
00-100
0-1000
00001
00010

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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