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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, C2.5C25, C4.13C24, D4.8C23, Q8.8C23, C22.3C24, C23.50C23, Q8(C2×D4), D4(C2×Q8), C4○D47C22, (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)]])

34 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2M 4A ··· 4T 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×Q8 C2×C4○D4 2- 1+4 C2 # reps 1 5 10 16 2

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

 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 0 2 0 0 3 3 2 4 0 2 0 0 0 0 3 0 2 2
,
 4 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 4 0 0 1 1 0 4
,
 4 0 0 0 0 0 0 1 0 0 0 4 0 0 0 0 4 4 1 2 0 0 1 4 4
,
 4 0 0 0 0 0 0 2 0 0 0 2 0 0 0 0 3 3 2 4 0 2 2 0 3

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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