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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, C2.5C25, C4.13C24, D4.8C23, Q8.8C23, C22.3C24, C23.50C23, Q8○(C2×D4), D4○(C2×Q8), C4○D4⋊7C22, (C2×C4).46C23, (C22×Q8)⋊10C2, (C2×Q8)⋊17C22, (C2×D4).82C22, (C22×C4).82C22, (C2×C4○D4)⋊14C2, SmallGroup(64,265)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2 — C2×2- 1+4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C22×Q8 — 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=1, d2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 417 in 397 conjugacy classes, 377 normal (4 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, D4, Q8, C23, C22×C4, C2×D4, C2×Q8, C4○D4, C22×Q8, C2×C4○D4, 2- 1+4, C2×2- 1+4
Quotients: C1, C2, C22, C23, C24, 2- 1+4, C25, C2×2- 1+4

Smallest permutation representation of C2×2- 1+4
►On 32 points
Generators in S32
(1 31)(2 32)(3 29)(4 30)(5 15)(6 16)(7 13)(8 14)(9 26)(10 27)(11 28)(12 25)(17 21)(18 22)(19 23)(20 24)

(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)

(2 4)(5 7)(10 12)(13 15)(18 20)(22 24)(25 27)(30 32)

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

(1 16 3 14)(2 13 4 15)(5 32 7 30)(6 29 8 31)(9 19 11 17)(10 20 12 18)(21 26 23 28)(22 27 24 25)
 
G:=sub<Sym(32)| (1,31)(2,32)(3,29)(4,30)(5,15)(6,16)(7,13)(8,14)(9,26)(10,27)(11,28)(12,25)(17,21)(18,22)(19,23)(20,24), (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), (2,4)(5,7)(10,12)(13,15)(18,20)(22,24)(25,27)(30,32), (1,28,3,26)(2,25,4,27)(5,18,7,20)(6,19,8,17)(9,31,11,29)(10,32,12,30)(13,24,15,22)(14,21,16,23), (1,16,3,14)(2,13,4,15)(5,32,7,30)(6,29,8,31)(9,19,11,17)(10,20,12,18)(21,26,23,28)(22,27,24,25)>;
 
G:=Group( (1,31)(2,32)(3,29)(4,30)(5,15)(6,16)(7,13)(8,14)(9,26)(10,27)(11,28)(12,25)(17,21)(18,22)(19,23)(20,24), (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), (2,4)(5,7)(10,12)(13,15)(18,20)(22,24)(25,27)(30,32), (1,28,3,26)(2,25,4,27)(5,18,7,20)(6,19,8,17)(9,31,11,29)(10,32,12,30)(13,24,15,22)(14,21,16,23), (1,16,3,14)(2,13,4,15)(5,32,7,30)(6,29,8,31)(9,19,11,17)(10,20,12,18)(21,26,23,28)(22,27,24,25) );
 
G=PermutationGroup([[(1,31),(2,32),(3,29),(4,30),(5,15),(6,16),(7,13),(8,14),(9,26),(10,27),(11,28),(12,25),(17,21),(18,22),(19,23),(20,24)], [(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)], [(2,4),(5,7),(10,12),(13,15),(18,20),(22,24),(25,27),(30,32)], [(1,28,3,26),(2,25,4,27),(5,18,7,20),(6,19,8,17),(9,31,11,29),(10,32,12,30),(13,24,15,22),(14,21,16,23)], [(1,16,3,14),(2,13,4,15),(5,32,7,30),(6,29,8,31),(9,19,11,17),(10,20,12,18),(21,26,23,28),(22,27,24,25)]])
 

C2×2- 1+4 is a maximal subgroup of
 2- 1+4⋊2C4  (C22×Q8)⋊C4  C23.4C24  M4(2).25C23  2- 1+4⋊4C4  (C2×Q8)⋊16D4  Q8.(C2×D4)  (C2×Q8)⋊17D4  M4(2).C23  C22.75C25  C22.76C25  C22.78C25  C4⋊2- 1+4  C22.88C25  C22.89C25  C4.C25  2- 1+6
C2×2- 1+4 is a maximal quotient of
 C22.50C25  C2×D4×Q8  C22.71C25  C22.75C25  C22.78C25  C22.81C25  C22.84C25  C4⋊2- 1+4  C22.88C25  C22.91C25  C22.92C25  C22.96C25  C22.98C25  C22.100C25  C22.104C25  C22.105C25  C22.107C25  C23.144C24  C22.111C25  C23.146C24  C22.120C25  C22.124C25  C22.125C25  C22.127C25  C22.130C25  C22.133C25  C22.136C25  C22.137C25  C22.139C25  C22.141C25  C22.142C25  C22.143C25  C22.144C25  C22.145C25  C22.146C25  C22.148C25  C22.150C25  C22.152C25  C22.153C25  C22.154C25

34 conjugacy classes

class 1 2A2B2C2D···2M4A···4T
order12222···24···4
size11112···22···2

34 irreducible representations

dim11114
type++++-
imageC1C2C2C22- 1+4
kernelC2×2- 1+4C22×Q8C2×C4○D42- 1+4C2
# reps1510162

Matrix representation of C2×2- 1+4 ►in GL5(𝔽5)

40000
04000
00400
00040
00004
,
40000
00020
03324
02000
03022
,
40000
01000
00100
00040
01104
,
40000
00100
04000
04412
00144
,
40000
00200
02000
03324
02203

G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,0,3,2,3,0,0,3,0,0,0,2,2,0,2,0,0,4,0,2],[4,0,0,0,0,0,1,0,0,1,0,0,1,0,1,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,4,0,0,1,0,4,1,0,0,0,1,4,0,0,0,2,4],[4,0,0,0,0,0,0,2,3,2,0,2,0,3,2,0,0,0,2,0,0,0,0,4,3] >;
 

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(64,265);
 
// by ID
 
G=gap.SmallGroup(64,265);
 
# by ID
 
G:=PCGroup([6,-2,2,2,2,2,-2,409,199,332,158,963]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^2=e^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^-1=b^2*d>;
 
// generators/relations
 

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