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G = A5⋊Q8  order 480 = 25·3·5

The semidirect product of A5 and Q8 acting via Q8/C4=C2

non-abelian, not soluble

Aliases: A5⋊Q8, C4.1S5, A5⋊C4.C2, C2.4(C2×S5), (C4×A5).2C2, (C2×A5).3C22, SmallGroup(480,945)

Series: Chief►Derived ►Lower central ►Upper central

Chief series
C1 — C2 — C4 — C4×A5 — A5⋊Q8
Derived series
A5 — C2×A5 — A5⋊Q8
Lower central
A5 — C2×A5 — A5⋊Q8
Upper central
C1 — C2 — C4

Subgroups: 724 in 74 conjugacy classes, 9 normal (7 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C2×C4, Q8, C23, D5, C10, Dic3, C12, A4, D6, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, F5, D10, Dic6, C4×S3, C3×Q8, C2×A4, C22⋊Q8, C4×D5, C2×F5, A4⋊C4, C4×A4, S3×Q8, A5, C4⋊F5, A4⋊Q8, C2×A5, A5⋊C4, C4×A5, A5⋊Q8
Quotients: C1, C2, C22, Q8, S5, C2×S5, A5⋊Q8

Character table of A5⋊Q8

 class 12A2B2C34A4B4C4D4E4F561012A12B12C20A20B
 size 11151520220203060602420244040402424
ρ11111111111111111111    trivial
ρ2111111-1-11-1-1111-11-111    linear of order 2
ρ311111-1-11-11-1111-1-11-1-1    linear of order 2
ρ411111-11-1-1-111111-1-1-1-1    linear of order 2
ρ52-2-2220000002-2-200000    symplectic lifted from Q8, Schur index 2
ρ644001-4-22000-11-11-1-111    orthogonal lifted from C2×S5
ρ744001-42-2000-11-1-1-1111    orthogonal lifted from C2×S5
ρ8440014-2-2000-11-1111-1-1    orthogonal lifted from S5
ρ944001422000-11-1-11-1-1-1    orthogonal lifted from S5
ρ105511-15111-1-10-101-1100    orthogonal lifted from S5
ρ115511-15-1-11110-10-1-1-100    orthogonal lifted from S5
ρ125511-1-51-1-11-10-1011-100    orthogonal lifted from C2×S5
ρ135511-1-5-11-1-110-10-11100    orthogonal lifted from C2×S5
ρ1466-2-20600-20010100011    orthogonal lifted from S5
ρ1566-2-20-600200101000-1-1    orthogonal lifted from C2×S5
ρ166-62-2000000010-1000-√-5√-5    complex faithful
ρ176-62-2000000010-1000√-5-√-5    complex faithful
ρ188-8002000000-2-2200000    symplectic faithful, Schur index 2
ρ1910-10-22-200000002000000    symplectic faithful, Schur index 2

Permutation representations of A5⋊Q8
►On 24 points - transitive group 24T1351
Generators in S24
(1 22 7 16)(2 14 8 20)(3 13 9 19)(4 23 10 17)(5 24 11 18)(6 15 12 21)

(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
 
G:=sub<Sym(24)| (1,22,7,16)(2,14,8,20)(3,13,9,19)(4,23,10,17)(5,24,11,18)(6,15,12,21), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)>;
 
G:=Group( (1,22,7,16)(2,14,8,20)(3,13,9,19)(4,23,10,17)(5,24,11,18)(6,15,12,21), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24) );
 
G=PermutationGroup([[(1,22,7,16),(2,14,8,20),(3,13,9,19),(4,23,10,17),(5,24,11,18),(6,15,12,21)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)]])
 
G:=TransitiveGroup(24,1351);
 

Matrix representation of A5⋊Q8 ►in GL6(𝔽3)

020002
100202
000210
000201
002201
000101
,
222100
102220
221202
212000
120100
022200

G:=sub<GL(6,GF(3))| [0,1,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,2,2,2,2,1,0,0,1,0,0,0,2,2,0,1,1,1],[2,1,2,2,1,0,2,0,2,1,2,2,2,2,1,2,0,2,1,2,2,0,1,2,0,2,0,0,0,0,0,0,2,0,0,0] >;
 

A5⋊Q8 in GAP, Magma, Sage, TeX 

A_5\rtimes Q_8
 
% in TeX
 
G:=Group("A5:Q8");
 
// GroupNames label
 
G:=SmallGroup(480,945);
 
// by ID
 
G=gap.SmallGroup(480,945);
 
# by ID
 

Export

Character table of A5⋊Q8 in TeX

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