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: Chief►Derived ►Lower central ►Upper central
Subgroups: 1370 in 120 conjugacy classes, 12 normal (8 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, D4, D4, C23, D5, C10, Dic3, C12, A4, D6, C2×C6, C22×C4, C2×D4, C24, Dic5, C20, D10, C2×C10, C4×S3, D12, C3⋊D4, C3×D4, C2×A4, C22×S3, C22×D4, C4×D5, D20, C5⋊D4, C5×D4, C22×D5, C4×A4, S3×D4, C22×A4, A5, D4×D5, D4×A4, C2×A5, C2×A5, C4×A5, C22×A5, D4×A5
Quotients: C1, C2, C22, D4, A5, C2×A5, C22×A5, D4×A5
Character table of D4×A5
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 5A | 5B | 6A | 6B | 6C | 10A | 10B | 10C | 10D | 10E | 10F | 12 | 20A | 20B | |
| size | 1 | 1 | 2 | 2 | 15 | 15 | 30 | 30 | 20 | 2 | 30 | 12 | 12 | 20 | 40 | 40 | 12 | 12 | 24 | 24 | 24 | 24 | 40 | 24 | 24 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ6 | 3 | 3 | -3 | -3 | -1 | -1 | 1 | 1 | 0 | 3 | -1 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | 0 | 1-√5/2 | 1+√5/2 | orthogonal lifted from C2×A5 |
| ρ7 | 3 | 3 | -3 | 3 | -1 | -1 | -1 | 1 | 0 | -3 | 1 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | 0 | -1+√5/2 | -1-√5/2 | orthogonal lifted from C2×A5 |
| ρ8 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 3 | -1 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 0 | 1+√5/2 | 1-√5/2 | orthogonal lifted from A5 |
| ρ9 | 3 | 3 | 3 | -3 | -1 | -1 | 1 | -1 | 0 | -3 | 1 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | -1+√5/2 | -1-√5/2 | 0 | -1+√5/2 | -1-√5/2 | orthogonal lifted from C2×A5 |
| ρ10 | 3 | 3 | -3 | 3 | -1 | -1 | -1 | 1 | 0 | -3 | 1 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | 0 | -1-√5/2 | -1+√5/2 | orthogonal lifted from C2×A5 |
| ρ11 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 3 | -1 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 0 | 1-√5/2 | 1+√5/2 | orthogonal lifted from A5 |
| ρ12 | 3 | 3 | 3 | -3 | -1 | -1 | 1 | -1 | 0 | -3 | 1 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | -1-√5/2 | -1+√5/2 | 0 | -1-√5/2 | -1+√5/2 | orthogonal lifted from C2×A5 |
| ρ13 | 3 | 3 | -3 | -3 | -1 | -1 | 1 | 1 | 0 | 3 | -1 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | 0 | 1+√5/2 | 1-√5/2 | orthogonal lifted from C2×A5 |
| ρ14 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 1 | 4 | 0 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | orthogonal lifted from A5 |
| ρ15 | 4 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 1 | -4 | 0 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | orthogonal lifted from C2×A5 |
| ρ16 | 4 | 4 | -4 | 4 | 0 | 0 | 0 | 0 | 1 | -4 | 0 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | orthogonal lifted from C2×A5 |
| ρ17 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 1 | 4 | 0 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from C2×A5 |
| ρ18 | 5 | 5 | -5 | -5 | 1 | 1 | -1 | -1 | -1 | 5 | 1 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | orthogonal lifted from C2×A5 |
| ρ19 | 5 | 5 | 5 | -5 | 1 | 1 | -1 | 1 | -1 | -5 | -1 | 0 | 0 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | orthogonal lifted from C2×A5 |
| ρ20 | 5 | 5 | -5 | 5 | 1 | 1 | 1 | -1 | -1 | -5 | -1 | 0 | 0 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | orthogonal lifted from C2×A5 |
| ρ21 | 5 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | -1 | 5 | 1 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | orthogonal lifted from A5 |
| ρ22 | 6 | -6 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 1+√5 | 1-√5 | 0 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ23 | 6 | -6 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 1-√5 | 1+√5 | 0 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ24 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ25 | 10 | -10 | 0 | 0 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 20 points - transitive group 20T119
Generators in S20
(1 9)(2 11)(3 5)(4 15)(6 13)(8 19)(10 17)(12 14)(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)(2,11)(3,5)(4,15)(6,13)(8,19)(10,17)(12,14)(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)(2,11)(3,5)(4,15)(6,13)(8,19)(10,17)(12,14)(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),(2,11),(3,5),(4,15),(6,13),(8,19),(10,17),(12,14),(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 6)(4 13)(5 16)(7 18)(8 20)(9 24)(10 15)(11 19)(12 21)(14 23)
(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)(4,13)(5,16)(7,18)(8,20)(9,24)(10,15)(11,19)(12,21)(14,23), (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)(4,13)(5,16)(7,18)(8,20)(9,24)(10,15)(11,19)(12,21)(14,23), (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),(4,13),(5,16),(7,18),(8,20),(9,24),(10,15),(11,19),(12,21),(14,23)], [(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 18)(2 7)(3 12)(4 23)(5 9)(10 20)(11 17)(13 19)(14 16)(15 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)| (1,18)(2,7)(3,12)(4,23)(5,9)(10,20)(11,17)(13,19)(14,16)(15,21), (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( (1,18)(2,7)(3,12)(4,23)(5,9)(10,20)(11,17)(13,19)(14,16)(15,21), (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([[(1,18),(2,7),(3,12),(4,23),(5,9),(10,20),(11,17),(13,19),(14,16),(15,21)], [(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(24,1344);
Matrix representation of D4×A5 ►in GL5(𝔽61)
| 60 | 2 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 22 | 48 |
| 0 | 0 | 30 | 9 | 39 |
| 1 | 0 | 0 | 0 | 0 |
| 1 | 60 | 0 | 0 | 0 |
| 0 | 0 | 48 | 52 | 39 |
| 0 | 0 | 39 | 31 | 52 |
| 0 | 0 | 0 | 1 | 0 |
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