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G = C2×C52⋊Q8  order 400 = 24·52

Direct product of C2 and C52⋊Q8

direct product, non-abelian, soluble, monomial, rational

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, C4, C22, C5, C2×C4, Q8, D5, C10, C2×Q8, F5, D10, C52, C2×F5, C5⋊D5, C5×C10, C52⋊C4, C2×C5⋊D5, C52⋊Q8, C2×C52⋊C4, C2×C52⋊Q8
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C52⋊Q8, C2×C52⋊Q8

Character table of C2×C52⋊Q8

 class 12A2B2C4A4B4C4D4E4F5A5B5C10A10B10C
 size 112525505050505050888888
ρ11111111111111111    trivial
ρ21111-1-111-1-1111111    linear of order 2
ρ311111-1-1-11-1111111    linear of order 2
ρ41111-11-1-1-11111111    linear of order 2
ρ51-11-1-1-11-111111-1-1-1    linear of order 2
ρ61-11-1111-1-1-1111-1-1-1    linear of order 2
ρ71-11-11-1-11-11111-1-1-1    linear of order 2
ρ81-11-1-11-111-1111-1-1-1    linear of order 2
ρ92-2-22000000222-2-2-2    symplectic lifted from Q8, Schur index 2
ρ1022-2-2000000222222    symplectic lifted from Q8, Schur index 2
ρ118-800000000-23-2-322    orthogonal faithful
ρ128800000000-2-23-23-2    orthogonal lifted from C52⋊Q8
ρ138-8000000003-2-222-3    orthogonal faithful
ρ148800000000-23-23-2-2    orthogonal lifted from C52⋊Q8
ρ158-800000000-2-232-32    orthogonal faithful
ρ1688000000003-2-2-2-23    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 16)(12 17)(13 18)(14 19)(15 20)

(11 12 13 14 15)(16 17 18 19 20)

(1 5 4 3 2)(6 10 9 8 7)

(1 14)(2 15 5 13)(3 11 4 12)(6 19)(7 20 10 18)(8 16 9 17)

(1 6)(2 9 5 8)(3 7 4 10)(11 18 12 20)(13 17 15 16)(14 19)
 
G:=sub<Sym(20)| (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20), (11,12,13,14,15)(16,17,18,19,20), (1,5,4,3,2)(6,10,9,8,7), (1,14)(2,15,5,13)(3,11,4,12)(6,19)(7,20,10,18)(8,16,9,17), (1,6)(2,9,5,8)(3,7,4,10)(11,18,12,20)(13,17,15,16)(14,19)>;
 
G:=Group( (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20), (11,12,13,14,15)(16,17,18,19,20), (1,5,4,3,2)(6,10,9,8,7), (1,14)(2,15,5,13)(3,11,4,12)(6,19)(7,20,10,18)(8,16,9,17), (1,6)(2,9,5,8)(3,7,4,10)(11,18,12,20)(13,17,15,16)(14,19) );
 
G=PermutationGroup([[(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20)], [(11,12,13,14,15),(16,17,18,19,20)], [(1,5,4,3,2),(6,10,9,8,7)], [(1,14),(2,15,5,13),(3,11,4,12),(6,19),(7,20,10,18),(8,16,9,17)], [(1,6),(2,9,5,8),(3,7,4,10),(11,18,12,20),(13,17,15,16),(14,19)]])
 
G:=TransitiveGroup(20,99);
 

Matrix representation of C2×C52⋊Q8 ►in GL8(ℤ)

-10000000
0-1000000
00-100000
000-10000
0000-1000
00000-100
000000-10
0000000-1
,
01000000
00100000
00010000
-1-1-1-10000
00000100
00000010
00000001
0000-1-1-1-1
,
01000000
00100000
00010000
-1-1-1-10000
0000-1-1-1-1
00001000
00000100
00000010
,
00001000
00000100
00000010
00000001
10000000
-1-1-1-10000
00010000
00100000
,
0000-1000
0000000-1
00000-100
00001111
-10000000
000-10000
0-1000000
11110000

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
 

Export

Character table of C2×C52⋊Q8 in TeX

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