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G = Q8.A5order 480 = 25·3·5

The non-split extension by Q8 of A5 acting through Inn(Q8)

non-abelian, not soluble

Aliases: Q8.A5, SL2(F5):4C22, C4.A5:2C2, C4.3(C2xA5), C2.6(C22xA5), SmallGroup(480,959)

Series: ChiefDerived Lower central Upper central

C1C2C4Q8 — Q8.A5
SL2(F5) — Q8.A5
SL2(F5) — Q8.A5
C1C2Q8

Subgroups: 924 in 85 conjugacy classes, 11 normal (5 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C2xC4, D4, Q8, Q8, C23, D5, C10, Dic3, C12, D6, C2xD4, C4oD4, Dic5, C20, D10, SL2(F3), C4xS3, D12, C3xQ8, 2+ 1+4, C4xD5, D20, C5xQ8, C4.A4, Q8:3S3, Q8:2D5, Q8.A4, SL2(F5), C4.A5, Q8.A5
Quotients: C1, C2, C22, A5, C2xA5, C22xA5, Q8.A5

Character table of Q8.A5

 class 12A2B2C2D34A4B4C4D5A5B610A10B12A12B12C20A20B20C20D20E20F
 size 1130303020222301212201212404040242424242424
ρ1111111111111111111111111    trivial
ρ211-1-1111-1-11111111-1-1-1-1-1-111    linear of order 2
ρ3111-1-11-1-11111111-11-1-1-111-1-1    linear of order 2
ρ411-11-11-11-1111111-1-1111-1-1-1-1    linear of order 2
ρ53311-103-3-3-11+5/21-5/201-5/21+5/2000-1-5/2-1+5/2-1-5/2-1+5/21+5/21-5/2    orthogonal lifted from C2xA5
ρ633-1-1-10333-11+5/21-5/201-5/21+5/20001+5/21-5/21+5/21-5/21+5/21-5/2    orthogonal lifted from A5
ρ733-1110-3-33-11-5/21+5/201+5/21-5/2000-1+5/2-1-5/21-5/21+5/2-1+5/2-1-5/2    orthogonal lifted from C2xA5
ρ8331-110-33-3-11-5/21+5/201+5/21-5/20001-5/21+5/2-1+5/2-1-5/2-1+5/2-1-5/2    orthogonal lifted from C2xA5
ρ93311-103-3-3-11-5/21+5/201+5/21-5/2000-1+5/2-1-5/2-1+5/2-1-5/21-5/21+5/2    orthogonal lifted from C2xA5
ρ1033-1110-3-33-11+5/21-5/201-5/21+5/2000-1-5/2-1+5/21+5/21-5/2-1-5/2-1+5/2    orthogonal lifted from C2xA5
ρ11331-110-33-3-11+5/21-5/201-5/21+5/20001+5/21-5/2-1-5/2-1+5/2-1-5/2-1+5/2    orthogonal lifted from C2xA5
ρ1233-1-1-10333-11-5/21+5/201+5/21-5/20001-5/21+5/21-5/21+5/21-5/21+5/2    orthogonal lifted from A5
ρ134400014-4-40-1-11-1-11-1-11111-1-1    orthogonal lifted from C2xA5
ρ14440001-4-440-1-11-1-1-11-111-1-111    orthogonal lifted from C2xA5
ρ15440001-44-40-1-11-1-1-1-11-1-11111    orthogonal lifted from C2xA5
ρ164400014440-1-11-1-1111-1-1-1-1-1-1    orthogonal lifted from A5
ρ174-4000-20000-1-5-1+521-51+5000000000    orthogonal faithful
ρ184-4000-20000-1+5-1-521+51-5000000000    orthogonal faithful
ρ1955-11-1-1-55-5100-10011-1000000    orthogonal lifted from C2xA5
ρ2055-1-11-15-5-5100-100-111000000    orthogonal lifted from C2xA5
ρ2155111-1555100-100-1-1-1000000    orthogonal lifted from A5
ρ22551-1-1-1-5-55100-1001-11000000    orthogonal lifted from C2xA5
ρ238-800020000-2-2-222000000000    orthogonal faithful, Schur index 2
ρ2412-1200000000220-2-2000000000    orthogonal faithful

Smallest permutation representation of Q8.A5
On 48 points
Generators in S48
(1 9 48 6 32 15 3 19 38 8 42 25)(2 24 33 5 37 10 4 14 43 7 47 20)(11 36 30 34 23 17 21 46 40 44 13 27)(12 26 41 45 39 18 22 16 31 35 29 28)
(1 2 3 4)(5 6 7 8)(9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28)(29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)

G:=sub<Sym(48)| (1,9,48,6,32,15,3,19,38,8,42,25)(2,24,33,5,37,10,4,14,43,7,47,20)(11,36,30,34,23,17,21,46,40,44,13,27)(12,26,41,45,39,18,22,16,31,35,29,28), (1,2,3,4)(5,6,7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)>;

G:=Group( (1,9,48,6,32,15,3,19,38,8,42,25)(2,24,33,5,37,10,4,14,43,7,47,20)(11,36,30,34,23,17,21,46,40,44,13,27)(12,26,41,45,39,18,22,16,31,35,29,28), (1,2,3,4)(5,6,7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48) );

G=PermutationGroup([[(1,9,48,6,32,15,3,19,38,8,42,25),(2,24,33,5,37,10,4,14,43,7,47,20),(11,36,30,34,23,17,21,46,40,44,13,27),(12,26,41,45,39,18,22,16,31,35,29,28)], [(1,2,3,4),(5,6,7,8),(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28),(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)]])

Matrix representation of Q8.A5 in GL4(F5) generated by

3300
2000
0032
0044
,
0001
0040
3400
2300
G:=sub<GL(4,GF(5))| [3,2,0,0,3,0,0,0,0,0,3,4,0,0,2,4],[0,0,3,2,0,0,4,3,0,4,0,0,1,0,0,0] >;

Q8.A5 in GAP, Magma, Sage, TeX

Q_8.A_5
% in TeX

G:=Group("Q8.A5");
// GroupNames label

G:=SmallGroup(480,959);
// by ID

G=gap.SmallGroup(480,959);
# by ID

Export

Character table of Q8.A5 in TeX

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