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## G = C2×C52⋊C8order 400 = 24·52

### Direct product of C2 and C52⋊C8

Aliases: C2×C52⋊C8, (C5×C10)⋊C8, C5⋊D52C8, C521(C2×C8), C5⋊F5.C4, C5⋊F5.3C22, (C2×C5⋊D5).3C4, C5⋊D5.3(C2×C4), (C2×C5⋊F5).4C2, SmallGroup(400,208)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C2×C52⋊C8
 Chief series C1 — C52 — C5⋊D5 — C5⋊F5 — C52⋊C8 — C2×C52⋊C8
 Lower central C52 — C2×C52⋊C8
 Upper central C1 — C2

Generators and relations for C2×C52⋊C8
G = < a,b,c,d | a2=b5=c5=d8=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=bc-1 >

25C2
25C2
2C5
2C5
2C5
25C22
25C4
25C4
2C10
2C10
2C10
10D5
10D5
10D5
10D5
10D5
10D5
25C2×C4
25C8
25C8
10D10
10D10
10F5
10F5
10D10
10F5
10F5
10F5
10F5
25C2×C8
10C2×F5
10C2×F5
10C2×F5

Character table of C2×C52⋊C8

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 5C 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C size 1 1 25 25 25 25 25 25 8 8 8 25 25 25 25 25 25 25 25 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 -1 1 -1 1 1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 -1 linear of order 2 ρ3 1 -1 -1 1 -1 1 1 -1 1 1 1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 linear of order 2 ρ5 1 1 1 1 -1 -1 -1 -1 1 1 1 i -i -i i i -i -i i 1 1 1 linear of order 4 ρ6 1 1 1 1 -1 -1 -1 -1 1 1 1 -i i i -i -i i i -i 1 1 1 linear of order 4 ρ7 1 -1 -1 1 1 -1 -1 1 1 1 1 -i -i -i i i i i -i -1 -1 -1 linear of order 4 ρ8 1 -1 -1 1 1 -1 -1 1 1 1 1 i i i -i -i -i -i i -1 -1 -1 linear of order 4 ρ9 1 -1 1 -1 i i -i -i 1 1 1 ζ83 ζ8 ζ85 ζ83 ζ87 ζ85 ζ8 ζ87 -1 -1 -1 linear of order 8 ρ10 1 -1 1 -1 -i -i i i 1 1 1 ζ8 ζ83 ζ87 ζ8 ζ85 ζ87 ζ83 ζ85 -1 -1 -1 linear of order 8 ρ11 1 -1 1 -1 -i -i i i 1 1 1 ζ85 ζ87 ζ83 ζ85 ζ8 ζ83 ζ87 ζ8 -1 -1 -1 linear of order 8 ρ12 1 -1 1 -1 i i -i -i 1 1 1 ζ87 ζ85 ζ8 ζ87 ζ83 ζ8 ζ85 ζ83 -1 -1 -1 linear of order 8 ρ13 1 1 -1 -1 -i i -i i 1 1 1 ζ87 ζ8 ζ85 ζ83 ζ87 ζ8 ζ85 ζ83 1 1 1 linear of order 8 ρ14 1 1 -1 -1 i -i i -i 1 1 1 ζ85 ζ83 ζ87 ζ8 ζ85 ζ83 ζ87 ζ8 1 1 1 linear of order 8 ρ15 1 1 -1 -1 i -i i -i 1 1 1 ζ8 ζ87 ζ83 ζ85 ζ8 ζ87 ζ83 ζ85 1 1 1 linear of order 8 ρ16 1 1 -1 -1 -i i -i i 1 1 1 ζ83 ζ85 ζ8 ζ87 ζ83 ζ85 ζ8 ζ87 1 1 1 linear of order 8 ρ17 8 -8 0 0 0 0 0 0 -2 3 -2 0 0 0 0 0 0 0 0 2 -3 2 orthogonal faithful ρ18 8 8 0 0 0 0 0 0 -2 -2 3 0 0 0 0 0 0 0 0 3 -2 -2 orthogonal lifted from C52⋊C8 ρ19 8 -8 0 0 0 0 0 0 3 -2 -2 0 0 0 0 0 0 0 0 2 2 -3 orthogonal faithful ρ20 8 8 0 0 0 0 0 0 -2 3 -2 0 0 0 0 0 0 0 0 -2 3 -2 orthogonal lifted from C52⋊C8 ρ21 8 8 0 0 0 0 0 0 3 -2 -2 0 0 0 0 0 0 0 0 -2 -2 3 orthogonal lifted from C52⋊C8 ρ22 8 -8 0 0 0 0 0 0 -2 -2 3 0 0 0 0 0 0 0 0 -3 2 2 orthogonal faithful

Permutation representations of C2×C52⋊C8
On 20 points - transitive group 20T110
Generators in S20
(1 3)(2 4)(5 16)(6 17)(7 18)(8 19)(9 20)(10 13)(11 14)(12 15)
(1 12 6 10 8)(2 9 11 7 5)(3 15 17 13 19)(4 20 14 18 16)
(1 8 10 6 12)(2 7 9 5 11)(3 19 13 17 15)(4 18 20 16 14)
(1 2)(3 4)(5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20)

G:=sub<Sym(20)| (1,3)(2,4)(5,16)(6,17)(7,18)(8,19)(9,20)(10,13)(11,14)(12,15), (1,12,6,10,8)(2,9,11,7,5)(3,15,17,13,19)(4,20,14,18,16), (1,8,10,6,12)(2,7,9,5,11)(3,19,13,17,15)(4,18,20,16,14), (1,2)(3,4)(5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20)>;

G:=Group( (1,3)(2,4)(5,16)(6,17)(7,18)(8,19)(9,20)(10,13)(11,14)(12,15), (1,12,6,10,8)(2,9,11,7,5)(3,15,17,13,19)(4,20,14,18,16), (1,8,10,6,12)(2,7,9,5,11)(3,19,13,17,15)(4,18,20,16,14), (1,2)(3,4)(5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20) );

G=PermutationGroup([[(1,3),(2,4),(5,16),(6,17),(7,18),(8,19),(9,20),(10,13),(11,14),(12,15)], [(1,12,6,10,8),(2,9,11,7,5),(3,15,17,13,19),(4,20,14,18,16)], [(1,8,10,6,12),(2,7,9,5,11),(3,19,13,17,15),(4,18,20,16,14)], [(1,2),(3,4),(5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20)]])

G:=TransitiveGroup(20,110);

On 20 points - transitive group 20T113
Generators in S20
(1 3)(2 4)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)
(1 11 13 17 7)(3 19 5 9 15)
(1 13 7 11 17)(2 8 18 14 12)(3 5 15 19 9)(4 16 10 6 20)
(1 2 3 4)(5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20)

G:=sub<Sym(20)| (1,3)(2,4)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20), (1,11,13,17,7)(3,19,5,9,15), (1,13,7,11,17)(2,8,18,14,12)(3,5,15,19,9)(4,16,10,6,20), (1,2,3,4)(5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20)>;

G:=Group( (1,3)(2,4)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20), (1,11,13,17,7)(3,19,5,9,15), (1,13,7,11,17)(2,8,18,14,12)(3,5,15,19,9)(4,16,10,6,20), (1,2,3,4)(5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20) );

G=PermutationGroup([[(1,3),(2,4),(5,13),(6,14),(7,15),(8,16),(9,17),(10,18),(11,19),(12,20)], [(1,11,13,17,7),(3,19,5,9,15)], [(1,13,7,11,17),(2,8,18,14,12),(3,5,15,19,9),(4,16,10,6,20)], [(1,2,3,4),(5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20)]])

G:=TransitiveGroup(20,113);

Matrix representation of C2×C52⋊C8 in GL8(ℤ)

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

G:=sub<GL(8,Integers())| [-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0],[0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1],[0,0,0,0,1,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0] >;

C2×C52⋊C8 in GAP, Magma, Sage, TeX

C_2\times C_5^2\rtimes C_8
% in TeX

G:=Group("C2xC5^2:C8");
// GroupNames label

G:=SmallGroup(400,208);
// by ID

G=gap.SmallGroup(400,208);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,5,24,50,10564,256,262,9797,1457,1463]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^5=c^5=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^2,d*c*d^-1=b*c^-1>;
// generators/relations

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