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## G = (C5×C10).Q8order 400 = 24·52

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

Aliases: (C5×C10).Q8, C5⋊D5.4D4, C524(C4⋊C4), C2.(C52⋊Q8), C52⋊C44C4, 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 >

25C2
25C2
2C5
2C5
2C5
25C4
25C22
25C4
50C4
50C4
2C10
2C10
2C10
10D5
10D5
10D5
10D5
10D5
10D5
25C2×C4
25C2×C4
25C2×C4
10F5
10F5
10F5
10F5
10F5
10F5
10D10
10D10
10D10
25C4⋊C4
10C2×F5
10C2×F5
10C2×F5

Character table of (C5×C10).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 -i -i i -1 1 i 1 1 1 -1 -1 -1 linear of order 4 ρ6 1 -1 1 -1 i i -i -1 1 -i 1 1 1 -1 -1 -1 linear of order 4 ρ7 1 -1 1 -1 i -i i 1 -1 -i 1 1 1 -1 -1 -1 linear of order 4 ρ8 1 -1 1 -1 -i i -i 1 -1 i 1 1 1 -1 -1 -1 linear of order 4 ρ9 2 -2 -2 2 0 0 0 0 0 0 2 2 2 -2 -2 -2 orthogonal lifted from D4 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 (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 -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 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 -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 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 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 -1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 -1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 1 0 0 0 0 0 0

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

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