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G = C2×C52⋊C8  order 400 = 24·52

Direct product of C2 and C52⋊C8

direct product, metabelian, soluble, monomial, A-group

Aliases: C2×C52⋊C8, (C5×C10)⋊C8, C5⋊D5⋊2C8, C52⋊1(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 >

C1
C2
25C2
25C2
2C5
2C5
2C5
25C22
25C4
25C4
2C10
2C10
2C10
10D5
10D5
10D5
10D5
10D5
10D5
C52
25C2×C4
25C8
25C8
10D10
10D10
10F5
10F5
10D10
10F5
10F5
10F5
10F5
C5⋊D5
C5⋊D5
C5×C10
25C2×C8
10C2×F5
10C2×F5
10C2×F5
C2×C5⋊D5
C5⋊F5
C5⋊F5
C2×C5⋊F5
C52⋊C8
C52⋊C8
C2×C52⋊C8

Character table of C2×C52⋊C8

 class 12A2B2C4A4B4C4D5A5B5C8A8B8C8D8E8F8G8H10A10B10C
 size 112525252525258882525252525252525888
ρ11111111111111111111111    trivial
ρ21-1-11-111-11111-1-1-1-1111-1-1-1    linear of order 2
ρ31-1-11-111-1111-11111-1-1-1-1-1-1    linear of order 2
ρ411111111111-1-1-1-1-1-1-1-1111    linear of order 2
ρ51111-1-1-1-1111i-i-iii-i-ii111    linear of order 4
ρ61111-1-1-1-1111-iii-i-iii-i111    linear of order 4
ρ71-1-111-1-11111-i-i-iiiii-i-1-1-1    linear of order 4
ρ81-1-111-1-11111iii-i-i-i-ii-1-1-1    linear of order 4
ρ91-11-1ii-i-i111ζ83ζ8ζ85ζ83ζ87ζ85ζ8ζ87-1-1-1    linear of order 8
ρ101-11-1-i-iii111ζ8ζ83ζ87ζ8ζ85ζ87ζ83ζ85-1-1-1    linear of order 8
ρ111-11-1-i-iii111ζ85ζ87ζ83ζ85ζ8ζ83ζ87ζ8-1-1-1    linear of order 8
ρ121-11-1ii-i-i111ζ87ζ85ζ8ζ87ζ83ζ8ζ85ζ83-1-1-1    linear of order 8
ρ1311-1-1-ii-ii111ζ87ζ8ζ85ζ83ζ87ζ8ζ85ζ83111    linear of order 8
ρ1411-1-1i-ii-i111ζ85ζ83ζ87ζ8ζ85ζ83ζ87ζ8111    linear of order 8
ρ1511-1-1i-ii-i111ζ8ζ87ζ83ζ85ζ8ζ87ζ83ζ85111    linear of order 8
ρ1611-1-1-ii-ii111ζ83ζ85ζ8ζ87ζ83ζ85ζ8ζ87111    linear of order 8
ρ178-8000000-23-2000000002-32    orthogonal faithful
ρ1888000000-2-23000000003-2-2    orthogonal lifted from C52⋊C8
ρ198-80000003-2-20000000022-3    orthogonal faithful
ρ2088000000-23-200000000-23-2    orthogonal lifted from C52⋊C8
ρ21880000003-2-200000000-2-23    orthogonal lifted from C52⋊C8
ρ228-8000000-2-2300000000-322    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)

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

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
 

Export

Subgroup lattice of C2×C52⋊C8 in TeX
Character table of C2×C52⋊C8 in TeX

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