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## G = C2×C52⋊Q8order 400 = 24·52

### Direct product of C2 and C52⋊Q8

Aliases: C2×C52⋊Q8, C5⋊D5⋊Q8, (C5×C10)⋊Q8, C52⋊(C2×Q8), C5⋊D5.4C23, C52⋊C4.2C22, (C2×C52⋊C4).6C2, (C2×C5⋊D5).11C22, SmallGroup(400,212)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C5⋊D5 — C2×C52⋊Q8
 Chief series C1 — C52 — C5⋊D5 — C52⋊C4 — C52⋊Q8 — C2×C52⋊Q8
 Lower central C52 — C5⋊D5 — C2×C52⋊Q8
 Upper central C1 — C2

Generators and relations for C2×C52⋊Q8
G = < a,b,c,d,e | a2=b5=c5=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=c-1, ebe-1=b3, dcd-1=b, ece-1=c2, ede-1=d-1 >

Subgroups: 638 in 62 conjugacy classes, 21 normal (8 characteristic)
C1, C2, C2 [×2], C4 [×6], C22, C5 [×3], C2×C4 [×3], Q8 [×4], D5 [×6], C10 [×3], C2×Q8, F5 [×6], D10 [×3], C52, C2×F5 [×3], C5⋊D5 [×2], C5×C10, C52⋊C4 [×6], C2×C5⋊D5, C52⋊Q8 [×4], C2×C52⋊C4 [×3], C2×C52⋊Q8
Quotients: C1, C2 [×7], C22 [×7], Q8 [×2], C23, C2×Q8, C52⋊Q8, C2×C52⋊Q8

Character table of C2×C52⋊Q8

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 5C 10A 10B 10C size 1 1 25 25 50 50 50 50 50 50 8 8 8 8 8 8 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 -1 1 -1 -1 -1 1 -1 1 1 1 1 1 -1 -1 -1 linear of order 2 ρ6 1 -1 1 -1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 linear of order 2 ρ7 1 -1 1 -1 1 -1 -1 1 -1 1 1 1 1 -1 -1 -1 linear of order 2 ρ8 1 -1 1 -1 -1 1 -1 1 1 -1 1 1 1 -1 -1 -1 linear of order 2 ρ9 2 -2 -2 2 0 0 0 0 0 0 2 2 2 -2 -2 -2 symplectic lifted from Q8, Schur index 2 ρ10 2 2 -2 -2 0 0 0 0 0 0 2 2 2 2 2 2 symplectic lifted from Q8, Schur index 2 ρ11 8 -8 0 0 0 0 0 0 0 0 -2 3 -2 -3 2 2 orthogonal faithful ρ12 8 8 0 0 0 0 0 0 0 0 -2 -2 3 -2 3 -2 orthogonal lifted from C52⋊Q8 ρ13 8 -8 0 0 0 0 0 0 0 0 3 -2 -2 2 2 -3 orthogonal faithful ρ14 8 8 0 0 0 0 0 0 0 0 -2 3 -2 3 -2 -2 orthogonal lifted from C52⋊Q8 ρ15 8 -8 0 0 0 0 0 0 0 0 -2 -2 3 2 -3 2 orthogonal faithful ρ16 8 8 0 0 0 0 0 0 0 0 3 -2 -2 -2 -2 3 orthogonal lifted from C52⋊Q8

Permutation representations of C2×C52⋊Q8
On 20 points - transitive group 20T99
Generators in S20
(1 6)(2 7)(3 8)(4 9)(5 10)(11 19)(12 20)(13 16)(14 17)(15 18)
(11 12 13 14 15)(16 17 18 19 20)
(1 5 4 3 2)(6 10 9 8 7)
(1 11)(2 12 5 15)(3 13 4 14)(6 19)(7 20 10 18)(8 16 9 17)
(1 6)(2 9 5 8)(3 7 4 10)(11 19)(12 16 15 17)(13 18 14 20)

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

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

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

G:=TransitiveGroup(20,99);

Matrix representation of C2×C52⋊Q8 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
,
 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 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 -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 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 -1 -1 -1 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0
,
 0 0 0 0 -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 -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],[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,-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,-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],[0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,-1,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,-1,0,1,-1,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,-1,0,1,0,0,0,0] >;

C2×C52⋊Q8 in GAP, Magma, Sage, TeX

C_2\times C_5^2\rtimes Q_8
% in TeX

G:=Group("C2xC5^2:Q8");
// GroupNames label

G:=SmallGroup(400,212);
// by ID

G=gap.SmallGroup(400,212);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,5,48,121,55,964,1210,262,8645,1163,1463]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^5=c^5=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=c^-1,e*b*e^-1=b^3,d*c*d^-1=b,e*c*e^-1=c^2,e*d*e^-1=d^-1>;
// generators/relations

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