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G = D4×A5order 480 = 25·3·5

Direct product of D4 and A5

direct product, non-abelian, not soluble

Aliases: D4×A5, C4⋊(C2×A5), C22⋊(C2×A5), (C4×A5)⋊1C2, (C22×A5)⋊1C2, C2.3(C22×A5), (C2×A5).7C22, SmallGroup(480,956)

Series: ChiefDerived Lower central Upper central

C1C2C22D4 — D4×A5
A5C2×A5 — D4×A5
A5C2×A5 — D4×A5
C1C2D4

Subgroups: 1370 in 120 conjugacy classes, 12 normal (8 characteristic)
C1, C2, C2 [×6], C3, C4, C4, C22 [×2], C22 [×13], C5, S3 [×4], C6 [×3], C2×C4 [×2], D4, D4 [×5], C23 [×9], D5 [×4], C10 [×3], Dic3, C12, A4, D6 [×7], C2×C6 [×2], C22×C4, C2×D4 [×4], C24 [×2], Dic5, C20, D10 [×7], C2×C10 [×2], C4×S3, D12, C3⋊D4 [×2], C3×D4, C2×A4 [×3], C22×S3 [×2], C22×D4, C4×D5, D20, C5⋊D4 [×2], C5×D4, C22×D5 [×2], C4×A4, S3×D4, C22×A4 [×2], A5, D4×D5, D4×A4, C2×A5, C2×A5 [×2], C4×A5, C22×A5 [×2], D4×A5
Quotients: C1, C2 [×3], C22, D4, A5, C2×A5 [×3], C22×A5, D4×A5

Character table of D4×A5

 class 12A2B2C2D2E2F2G34A4B5A5B6A6B6C10A10B10C10D10E10F1220A20B
 size 112215153030202301212204040121224242424402424
ρ11111111111111111111111111    trivial
ρ2111-111-111-1-1111-111111-1-1-1-1-1    linear of order 2
ρ311-1-111-1-1111111-1-111-1-1-1-1111    linear of order 2
ρ411-11111-11-1-11111-111-1-111-1-1-1    linear of order 2
ρ52-200-220020022-200-2-20000000    orthogonal lifted from D4
ρ633-3-3-1-11103-11-5/21+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
ρ733-33-1-1-110-311-5/21+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
ρ83333-1-1-1-103-11+5/21-5/20001-5/21+5/21-5/21+5/21+5/21-5/201+5/21-5/2    orthogonal lifted from A5
ρ9333-3-1-11-10-311-5/21+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
ρ1033-33-1-1-110-311+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
ρ113333-1-1-1-103-11-5/21+5/20001+5/21-5/21+5/21-5/21-5/21+5/201-5/21+5/2    orthogonal lifted from A5
ρ12333-3-1-11-10-311+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
ρ1333-3-3-1-11103-11+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
ρ1444440000140-1-1111-1-1-1-1-1-11-1-1    orthogonal lifted from A5
ρ15444-400001-40-1-11-11-1-1-1-111-111    orthogonal lifted from C2×A5
ρ1644-4400001-40-1-111-1-1-111-1-1-111    orthogonal lifted from C2×A5
ρ1744-4-40000140-1-11-1-1-1-111111-1-1    orthogonal lifted from C2×A5
ρ1855-5-511-1-1-15100-111000000-100    orthogonal lifted from C2×A5
ρ19555-511-11-1-5-100-11-1000000100    orthogonal lifted from C2×A5
ρ2055-55111-1-1-5-100-1-11000000100    orthogonal lifted from C2×A5
ρ2155551111-15100-1-1-1000000-100    orthogonal lifted from A5
ρ226-6002-2000001+51-5000-1+5-1-50000000    orthogonal faithful
ρ236-6002-2000001-51+5000-1-5-1+50000000    orthogonal faithful
ρ248-8000000200-2-2-200220000000    orthogonal faithful
ρ2510-1000-2200-20000200000000000    orthogonal faithful

Permutation representations of D4×A5
On 20 points - transitive group 20T119
Generators in S20
(1 9)(2 19)(4 15)(5 7)(6 13)(8 17)(10 11)(14 16)(18 20)
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)

G:=sub<Sym(20)| (1,9)(2,19)(4,15)(5,7)(6,13)(8,17)(10,11)(14,16)(18,20), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)>;

G:=Group( (1,9)(2,19)(4,15)(5,7)(6,13)(8,17)(10,11)(14,16)(18,20), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20) );

G=PermutationGroup([(1,9),(2,19),(4,15),(5,7),(6,13),(8,17),(10,11),(14,16),(18,20)], [(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20)])

G:=TransitiveGroup(20,119);

On 24 points - transitive group 24T1343
Generators in S24
(1 17)(2 22)(3 8)(4 13)(5 15)(6 19)(7 21)(9 23)(10 20)(11 24)(12 16)(14 18)
(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)| (1,17)(2,22)(3,8)(4,13)(5,15)(6,19)(7,21)(9,23)(10,20)(11,24)(12,16)(14,18), (3,4)(5,6,7,8,9)(10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24)>;

G:=Group( (1,17)(2,22)(3,8)(4,13)(5,15)(6,19)(7,21)(9,23)(10,20)(11,24)(12,16)(14,18), (3,4)(5,6,7,8,9)(10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24) );

G=PermutationGroup([(1,17),(2,22),(3,8),(4,13),(5,15),(6,19),(7,21),(9,23),(10,20),(11,24),(12,16),(14,18)], [(3,4),(5,6,7,8,9),(10,11,12,13,14),(15,16,17,18,19,20,21,22,23,24)])

G:=TransitiveGroup(24,1343);

On 24 points - transitive group 24T1344
Generators in S24
(1 17)(2 22)(3 6)(4 14)(5 16)(7 18)(8 15)(9 19)(10 23)(11 20)(12 24)(13 21)
(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)| (1,17)(2,22)(3,6)(4,14)(5,16)(7,18)(8,15)(9,19)(10,23)(11,20)(12,24)(13,21), (3,4)(5,6,7,8,9)(10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24)>;

G:=Group( (1,17)(2,22)(3,6)(4,14)(5,16)(7,18)(8,15)(9,19)(10,23)(11,20)(12,24)(13,21), (3,4)(5,6,7,8,9)(10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24) );

G=PermutationGroup([(1,17),(2,22),(3,6),(4,14),(5,16),(7,18),(8,15),(9,19),(10,23),(11,20),(12,24),(13,21)], [(3,4),(5,6,7,8,9),(10,11,12,13,14),(15,16,17,18,19,20,21,22,23,24)])

G:=TransitiveGroup(24,1344);

Matrix representation of D4×A5 in GL5(𝔽61)

602000
01000
00100
0092248
0030939
,
10000
160000
00485239
00393152
00010

G:=sub<GL(5,GF(61))| [60,0,0,0,0,2,1,0,0,0,0,0,1,9,30,0,0,0,22,9,0,0,0,48,39],[1,1,0,0,0,0,60,0,0,0,0,0,48,39,0,0,0,52,31,1,0,0,39,52,0] >;

D4×A5 in GAP, Magma, Sage, TeX

D_4\times A_5
% in TeX

G:=Group("D4xA5");
// GroupNames label

G:=SmallGroup(480,956);
// by ID

G=gap.SmallGroup(480,956);
# by ID

Export

Character table of D4×A5 in TeX

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