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G = D4.A5order 480 = 25·3·5

The non-split extension by D4 of A5 acting through Inn(D4)

non-abelian, not soluble

Aliases: D4.A5, SL2(𝔽5).5C22, C4.A51C2, C4.1(C2×A5), C22.(C2×A5), C2.4(C22×A5), (C2×SL2(𝔽5))⋊1C2, SmallGroup(480,957)

Series: ChiefDerived Lower central Upper central

C1C2C22D4 — D4.A5
SL2(𝔽5) — D4.A5
SL2(𝔽5) — D4.A5
C1C2D4

Subgroups: 748 in 83 conjugacy classes, 11 normal (7 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, D4, D4, Q8, D5, C10, Dic3, C12, D6, C2×C6, C2×Q8, C4○D4, Dic5, C20, D10, C2×C10, SL2(𝔽3), Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, 2- 1+4, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C2×SL2(𝔽3), C4.A4, D42S3, D42D5, D4.A4, SL2(𝔽5), C4.A5, C2×SL2(𝔽5), D4.A5
Quotients: C1, C2, C22, A5, C2×A5, C22×A5, D4.A5

Character table of D4.A5

 class 12A2B2C2D34A4B4C4D5A5B6A6B6C10A10B10C10D10E10F1220A20B
 size 1122302023030301212204040121224242424402424
ρ1111111111111111111111111    trivial
ρ211-1-1111-1-11111-1-111-1-1-1-1111    linear of order 2
ρ3111-1-11-11-11111-111111-1-1-1-1-1    linear of order 2
ρ411-11-11-1-1111111-111-1-111-1-1-1    linear of order 2
ρ533-3310-31-1-11-5/21+5/20001-5/21+5/2-1-5/2-1+5/21+5/21-5/20-1-5/2-1+5/2    orthogonal lifted from C2×A5
ρ6333-310-3-11-11-5/21+5/20001-5/21+5/21+5/21-5/2-1-5/2-1+5/20-1-5/2-1+5/2    orthogonal lifted from C2×A5
ρ73333-103-1-1-11+5/21-5/20001+5/21-5/21-5/21+5/21-5/21+5/201-5/21+5/2    orthogonal lifted from A5
ρ833-3-3-10311-11-5/21+5/20001-5/21+5/2-1-5/2-1+5/2-1-5/2-1+5/201+5/21-5/2    orthogonal lifted from C2×A5
ρ9333-310-3-11-11+5/21-5/20001+5/21-5/21-5/21+5/2-1+5/2-1-5/20-1+5/2-1-5/2    orthogonal lifted from C2×A5
ρ103333-103-1-1-11-5/21+5/20001-5/21+5/21+5/21-5/21+5/21-5/201+5/21-5/2    orthogonal lifted from A5
ρ1133-3-3-10311-11+5/21-5/20001+5/21-5/2-1+5/2-1-5/2-1+5/2-1-5/201-5/21+5/2    orthogonal lifted from C2×A5
ρ1233-3310-31-1-11+5/21-5/20001+5/21-5/2-1+5/2-1-5/21-5/21+5/20-1+5/2-1-5/2    orthogonal lifted from C2×A5
ρ1344-4401-4000-1-111-1-1-111-1-1-111    orthogonal lifted from C2×A5
ρ14444-401-4000-1-11-11-1-1-1-111-111    orthogonal lifted from C2×A5
ρ154444014000-1-1111-1-1-1-1-1-11-1-1    orthogonal lifted from A5
ρ1644-4-4014000-1-11-1-1-1-111111-1-1    orthogonal lifted from C2×A5
ρ174-4000-20000-1+5-1-52001-51+50000000    symplectic faithful, Schur index 2
ρ184-4000-20000-1-5-1+52001+51-50000000    symplectic faithful, Schur index 2
ρ1955-55-1-1-5-11100-1-11000000100    orthogonal lifted from C2×A5
ρ20555-5-1-1-51-1100-11-1000000100    orthogonal lifted from C2×A5
ρ2155-5-51-15-1-1100-111000000-100    orthogonal lifted from C2×A5
ρ2255551-1511100-1-1-1000000-100    orthogonal lifted from A5
ρ238-800020000-2-2-200220000000    symplectic faithful, Schur index 2
ρ2412-120000000022000-2-20000000    symplectic faithful, Schur index 2

Smallest permutation representation of D4.A5
On 48 points
Generators in S48
(1 24 46 20 11)(2 10 15 43 23 8 4 21 37 17)(5 7 18 40 14)(6 12)(13 47 27 35 45 19 41 33 29 39)(16 44 30 32 48)(22 38 36 26 42)(25 31)
(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)| (1,24,46,20,11)(2,10,15,43,23,8,4,21,37,17)(5,7,18,40,14)(6,12)(13,47,27,35,45,19,41,33,29,39)(16,44,30,32,48)(22,38,36,26,42)(25,31), (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( (1,24,46,20,11)(2,10,15,43,23,8,4,21,37,17)(5,7,18,40,14)(6,12)(13,47,27,35,45,19,41,33,29,39)(16,44,30,32,48)(22,38,36,26,42)(25,31), (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([[(1,24,46,20,11),(2,10,15,43,23,8,4,21,37,17),(5,7,18,40,14),(6,12),(13,47,27,35,45,19,41,33,29,39),(16,44,30,32,48),(22,38,36,26,42),(25,31)], [(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 D4.A5 in GL4(𝔽5) generated by

0102
4020
0404
1020
,
2040
0401
4000
0204
G:=sub<GL(4,GF(5))| [0,4,0,1,1,0,4,0,0,2,0,2,2,0,4,0],[2,0,4,0,0,4,0,2,4,0,0,0,0,1,0,4] >;

D4.A5 in GAP, Magma, Sage, TeX

D_4.A_5
% in TeX

G:=Group("D4.A5");
// GroupNames label

G:=SmallGroup(480,957);
// by ID

G=gap.SmallGroup(480,957);
# by ID

Export

Character table of D4.A5 in TeX

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