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G = (C5×C10).Q8  order 400 = 24·52

The non-split extension by C5×C10 of Q8 acting faithfully

non-abelian, soluble, monomial

Aliases: (C5×C10).Q8, C5⋊D5.4D4, C52⋊4(C4⋊C4), C2.(C52⋊Q8), C52⋊C4⋊4C4, C5⋊D5.7(C2×C4), (C2×C52⋊C4).4C2, (C2×C5⋊D5).8C22, SmallGroup(400,134)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C52 — C5⋊D5 — (C5×C10).Q8
Chief series
C1 — C52 — C5⋊D5 — C2×C5⋊D5 — C2×C52⋊C4 — (C5×C10).Q8
Lower central
C52 — C5⋊D5 — (C5×C10).Q8
Upper central
C1 — C2

Generators and relations for (C5×C10).Q8
 G = < a,b,c,d | a5=b10=c4=1, d2=b5c2, ab=ba, cac-1=b8, dad-1=a3, cbc-1=a3b5, dbd-1=b7, dcd-1=b5c-1 >

C1
C2
25C2
25C2
2C5
2C5
2C5
25C4
25C22
25C4
50C4
50C4
2C10
2C10
2C10
10D5
10D5
10D5
10D5
10D5
10D5
C52
25C2×C4
25C2×C4
25C2×C4
10F5
10F5
10F5
10F5
10F5
10F5
10D10
10D10
10D10
C5⋊D5
C5⋊D5
C5×C10
25C4⋊C4
10C2×F5
10C2×F5
10C2×F5
C52⋊C4
C52⋊C4
C2×C5⋊D5
2C52⋊C4
2C52⋊C4
C2×C52⋊C4
C2×C52⋊C4
C2×C52⋊C4
(C5×C10).Q8

Character table of (C5×C10).Q8

 class 12A2B2C4A4B4C4D4E4F5A5B5C10A10B10C
 size 112525505050505050888888
ρ11111111111111111    trivial
ρ21111-1-1-111-1111111    linear of order 2
ρ31111-111-1-1-1111111    linear of order 2
ρ411111-1-1-1-11111111    linear of order 2
ρ51-11-1-i-ii-11i111-1-1-1    linear of order 4
ρ61-11-1ii-i-11-i111-1-1-1    linear of order 4
ρ71-11-1i-ii1-1-i111-1-1-1    linear of order 4
ρ81-11-1-ii-i1-1i111-1-1-1    linear of order 4
ρ92-2-22000000222-2-2-2    orthogonal lifted from D4
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 (C5×C10).Q8
►On 20 points - transitive group 20T111
Generators in S20
(1 5 7 3 9)(2 6 8 4 10)

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

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

(1 8 5 10)(2 7 6 9)(3 4)(11 16)(12 19 20 13)(14 15 18 17)
 
G:=sub<Sym(20)| (1,5,7,3,9)(2,6,8,4,10), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12,13,14,15,16,17,18,19,20), (1,20,6,17)(2,15,5,12)(3,16,4,11)(7,14,10,13)(8,19,9,18), (1,8,5,10)(2,7,6,9)(3,4)(11,16)(12,19,20,13)(14,15,18,17)>;
 
G:=Group( (1,5,7,3,9)(2,6,8,4,10), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12,13,14,15,16,17,18,19,20), (1,20,6,17)(2,15,5,12)(3,16,4,11)(7,14,10,13)(8,19,9,18), (1,8,5,10)(2,7,6,9)(3,4)(11,16)(12,19,20,13)(14,15,18,17) );
 
G=PermutationGroup([[(1,5,7,3,9),(2,6,8,4,10)], [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12,13,14,15,16,17,18,19,20)], [(1,20,6,17),(2,15,5,12),(3,16,4,11),(7,14,10,13),(8,19,9,18)], [(1,8,5,10),(2,7,6,9),(3,4),(11,16),(12,19,20,13),(14,15,18,17)]])
 
G:=TransitiveGroup(20,111);
 

►On 20 points - transitive group 20T112
Generators in S20
(1 3 5 7 9)(2 4 6 8 10)(11 19 17 15 13)(12 20 18 16 14)

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

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

(1 17 6 12)(2 20 5 19)(3 13 4 16)(7 15 10 14)(8 18 9 11)
 
G:=sub<Sym(20)| (1,3,5,7,9)(2,4,6,8,10)(11,19,17,15,13)(12,20,18,16,14), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,17)(2,18,10,16)(3,19,9,15)(4,20,8,14)(5,11,7,13)(6,12), (1,17,6,12)(2,20,5,19)(3,13,4,16)(7,15,10,14)(8,18,9,11)>;
 
G:=Group( (1,3,5,7,9)(2,4,6,8,10)(11,19,17,15,13)(12,20,18,16,14), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,17)(2,18,10,16)(3,19,9,15)(4,20,8,14)(5,11,7,13)(6,12), (1,17,6,12)(2,20,5,19)(3,13,4,16)(7,15,10,14)(8,18,9,11) );
 
G=PermutationGroup([[(1,3,5,7,9),(2,4,6,8,10),(11,19,17,15,13),(12,20,18,16,14)], [(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20)], [(1,17),(2,18,10,16),(3,19,9,15),(4,20,8,14),(5,11,7,13),(6,12)], [(1,17,6,12),(2,20,5,19),(3,13,4,16),(7,15,10,14),(8,18,9,11)]])
 
G:=TransitiveGroup(20,112);
 

Matrix representation of (C5×C10).Q8 ►in GL8(ℤ)

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

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

(C5×C10).Q8 in GAP, Magma, Sage, TeX 

(C_5\times C_{10}).Q_8
 
% in TeX
 
G:=Group("(C5xC10).Q8");
 
// GroupNames label
 
G:=SmallGroup(400,134);
 
// by ID
 
G=gap.SmallGroup(400,134);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-5,5,24,73,79,964,1210,496,8645,1163,2897]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^5=b^10=c^4=1,d^2=b^5*c^2,a*b=b*a,c*a*c^-1=b^8,d*a*d^-1=a^3,c*b*c^-1=a^3*b^5,d*b*d^-1=b^7,d*c*d^-1=b^5*c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of (C5×C10).Q8 in TeX
Character table of (C5×C10).Q8 in TeX

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