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G = (C5xC10).Q8  order 400 = 24·52

The non-split extension by C5xC10 of Q8 acting faithfully

non-abelian, soluble, monomial

Aliases: (C5xC10).Q8, C5:D5.4D4, C52:4(C4:C4), C2.(C52:Q8), C52:C4:4C4, C5:D5.7(C2xC4), (C2xC52:C4).4C2, (C2xC5:D5).8C22, SmallGroup(400,134)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C52 — C5:D5 — (C5xC10).Q8
Chief series
C1 — C52 — C5:D5 — C2xC5:D5 — C2xC52:C4 — (C5xC10).Q8
Lower central
C52 — C5:D5 — (C5xC10).Q8
Upper central
C1 — C2

Generators and relations for (C5xC10).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 >

Subgroups: 534 in 50 conjugacy classes, 13 normal (9 characteristic)
Quotients: C1, C2, C4, C22, C2xC4, D4, Q8, C4:C4, C52:Q8, (C5xC10).Q8
C1
C2
25C2
25C2
2C5
2C5
2C5
25C4
25C22
25C4
50C4
50C4
2C10
2C10
2C10
10D5
10D5
10D5
10D5
10D5
10D5
C52
25C2xC4
25C2xC4
25C2xC4
10F5
10F5
10F5
10F5
10F5
10F5
10D10
10D10
10D10
C5:D5
C5:D5
C5xC10
25C4:C4
10C2xF5
10C2xF5
10C2xF5
C52:C4
C52:C4
C2xC5:D5
2C52:C4
2C52:C4
C2xC52:C4
C2xC52:C4
C2xC52:C4
(C5xC10).Q8

Character table of (C5xC10).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 (C5xC10).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 (C5xC10).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] >;
 

(C5xC10).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 (C5xC10).Q8 in TeX
Character table of (C5xC10).Q8 in TeX

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