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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.A52C2, C4.3(C2×A5), C2.6(C22×A5), 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 [×3], C3, C4 [×3], C4, C22 [×5], C5, S3 [×3], C6, C2×C4 [×3], D4 [×6], Q8, Q8, C23 [×2], D5 [×3], C10, Dic3, C12 [×3], D6 [×3], C2×D4 [×3], C4○D4 [×4], Dic5, C20 [×3], D10 [×3], SL2(𝔽3), C4×S3 [×3], D12 [×3], C3×Q8, 2+ 1+4, C4×D5 [×3], D20 [×3], C5×Q8, C4.A4 [×3], Q83S3, Q82D5, Q8.A4, SL2(𝔽5), C4.A5 [×3], Q8.A5
Quotients: C1, C2 [×3], C22, A5, C2×A5 [×3], C22×A5, 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 C2×A5
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 C2×A5
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 C2×A5
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 C2×A5
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 C2×A5
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 C2×A5
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 C2×A5
ρ14440001-4-440-1-11-1-1-11-111-1-111    orthogonal lifted from C2×A5
ρ15440001-44-40-1-11-1-1-1-11-1-11111    orthogonal lifted from C2×A5
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 C2×A5
ρ2055-1-11-15-5-5100-100-111000000    orthogonal lifted from C2×A5
ρ2155111-1555100-100-1-1-1000000    orthogonal lifted from A5
ρ22551-1-1-1-5-55100-1001-11000000    orthogonal lifted from C2×A5
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 18 45 5 29 24 3 28 35 7 39 14)(2 13 30 8 34 19 4 23 40 6 44 9)(10 43 37 41 22 16 20 33 47 31 12 26)(11 25 48 32 46 17 21 15 38 42 36 27)
(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)| (1,18,45,5,29,24,3,28,35,7,39,14)(2,13,30,8,34,19,4,23,40,6,44,9)(10,43,37,41,22,16,20,33,47,31,12,26)(11,25,48,32,46,17,21,15,38,42,36,27), (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( (1,18,45,5,29,24,3,28,35,7,39,14)(2,13,30,8,34,19,4,23,40,6,44,9)(10,43,37,41,22,16,20,33,47,31,12,26)(11,25,48,32,46,17,21,15,38,42,36,27), (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([(1,18,45,5,29,24,3,28,35,7,39,14),(2,13,30,8,34,19,4,23,40,6,44,9),(10,43,37,41,22,16,20,33,47,31,12,26),(11,25,48,32,46,17,21,15,38,42,36,27)], [(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

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Character table of Q8.A5 in TeX

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