direct product, non-abelian, soluble, monomial
Aliases: C2×A4⋊D4, C23⋊4S4, C25⋊1S3, C24⋊3D6, (C2×A4)⋊2D4, A4⋊3(C2×D4), C22⋊3(C2×S4), A4⋊C4⋊2C22, (C23×A4)⋊2C2, (C22×S4)⋊3C2, (C2×S4)⋊3C22, C23⋊3(C3⋊D4), C2.21(C22×S4), (C2×A4).10C23, (C22×A4)⋊3C22, C23.10(C22×S3), (C2×A4⋊C4)⋊4C2, C22⋊(C2×C3⋊D4), SmallGroup(192,1488)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×A4⋊D4
G = < a,b,c,d,e,f | a2=b2=c2=d3=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, dbd-1=ebe-1=fbf=bc=cb, dcd-1=b, ce=ec, cf=fc, ede-1=fdf=d-1, fef=e-1 >
Subgroups: 1134 in 271 conjugacy classes, 35 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C2×C4, D4, C23, C23, C23, Dic3, A4, D6, C2×C6, C22⋊C4, C22×C4, C2×D4, C24, C24, C24, C2×Dic3, C3⋊D4, S4, C2×A4, C2×A4, C2×A4, C22×S3, C22×C6, C2×C22⋊C4, C22≀C2, C22×D4, C25, A4⋊C4, C2×C3⋊D4, C2×S4, C2×S4, C22×A4, C22×A4, C22×A4, C2×C22≀C2, C2×A4⋊C4, A4⋊D4, C22×S4, C23×A4, C2×A4⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, S4, C22×S3, C2×C3⋊D4, C2×S4, A4⋊D4, C22×S4, C2×A4⋊D4
Character table of C2×A4⋊D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 12 | 12 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
| 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 | 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 | 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 | 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 | 1 | 1 | -1 | linear of order 2 |
| ρ5 | 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 | 1 | -1 | 1 | linear of order 2 |
| ρ6 | 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 | 1 | 1 | 1 | linear of order 2 |
| ρ7 | 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 | 1 | -1 | -1 | linear of order 2 |
| ρ8 | 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 | 1 | 1 | -1 | linear of order 2 |
| ρ9 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | -2 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ14 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | -2 | 0 | 2 | orthogonal lifted from D4 |
| ρ15 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -√-3 | √-3 | 1 | √-3 | 1 | -√-3 | -1 | complex lifted from C3⋊D4 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | √-3 | √-3 | -1 | -√-3 | 1 | -√-3 | 1 | complex lifted from C3⋊D4 |
| ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | √-3 | -√-3 | 1 | -√-3 | 1 | √-3 | -1 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -√-3 | -√-3 | -1 | √-3 | 1 | √-3 | 1 | complex lifted from C3⋊D4 |
| ρ19 | 3 | -3 | -3 | 3 | 3 | -3 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 0 | 1 | 1 | -1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ20 | 3 | -3 | -3 | 3 | -3 | 3 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 0 | -1 | -1 | 1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ21 | 3 | 3 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 1 | -1 | 1 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ22 | 3 | 3 | 3 | 3 | -3 | -3 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 0 | -1 | 1 | -1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ23 | 3 | -3 | -3 | 3 | 3 | -3 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 0 | -1 | -1 | 1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ24 | 3 | -3 | -3 | 3 | -3 | 3 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 0 | 1 | 1 | -1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ25 | 3 | 3 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 0 | -1 | 1 | -1 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ26 | 3 | 3 | 3 | 3 | -3 | -3 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 0 | 1 | -1 | 1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ27 | 6 | -6 | 6 | -6 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4⋊D4 |
| ρ28 | 6 | 6 | -6 | -6 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4⋊D4 |
►On 24 points - transitive group 24T398
Generators in S24
(1 8)(2 5)(3 6)(4 7)(9 16)(10 13)(11 14)(12 15)(17 21)(18 22)(19 23)(20 24)
(1 8)(2 5)(3 6)(4 7)(9 16)(11 14)(17 21)(19 23)
(9 16)(10 13)(11 14)(12 15)(17 21)(18 22)(19 23)(20 24)
(1 11 22)(2 23 12)(3 9 24)(4 21 10)(5 19 15)(6 16 20)(7 17 13)(8 14 18)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(2 4)(5 7)(9 24)(10 23)(11 22)(12 21)(13 19)(14 18)(15 17)(16 20)
G:=sub<Sym(24)| (1,8)(2,5)(3,6)(4,7)(9,16)(10,13)(11,14)(12,15)(17,21)(18,22)(19,23)(20,24), (1,8)(2,5)(3,6)(4,7)(9,16)(11,14)(17,21)(19,23), (9,16)(10,13)(11,14)(12,15)(17,21)(18,22)(19,23)(20,24), (1,11,22)(2,23,12)(3,9,24)(4,21,10)(5,19,15)(6,16,20)(7,17,13)(8,14,18), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (2,4)(5,7)(9,24)(10,23)(11,22)(12,21)(13,19)(14,18)(15,17)(16,20)>;
G:=Group( (1,8)(2,5)(3,6)(4,7)(9,16)(10,13)(11,14)(12,15)(17,21)(18,22)(19,23)(20,24), (1,8)(2,5)(3,6)(4,7)(9,16)(11,14)(17,21)(19,23), (9,16)(10,13)(11,14)(12,15)(17,21)(18,22)(19,23)(20,24), (1,11,22)(2,23,12)(3,9,24)(4,21,10)(5,19,15)(6,16,20)(7,17,13)(8,14,18), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (2,4)(5,7)(9,24)(10,23)(11,22)(12,21)(13,19)(14,18)(15,17)(16,20) );
G=PermutationGroup([[(1,8),(2,5),(3,6),(4,7),(9,16),(10,13),(11,14),(12,15),(17,21),(18,22),(19,23),(20,24)], [(1,8),(2,5),(3,6),(4,7),(9,16),(11,14),(17,21),(19,23)], [(9,16),(10,13),(11,14),(12,15),(17,21),(18,22),(19,23),(20,24)], [(1,11,22),(2,23,12),(3,9,24),(4,21,10),(5,19,15),(6,16,20),(7,17,13),(8,14,18)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(2,4),(5,7),(9,24),(10,23),(11,22),(12,21),(13,19),(14,18),(15,17),(16,20)]])
G:=TransitiveGroup(24,398);
►On 24 points - transitive group 24T399
Generators in S24
(1 13)(2 14)(3 15)(4 16)(5 22)(6 23)(7 24)(8 21)(9 19)(10 20)(11 17)(12 18)
(1 2)(3 4)(5 7)(6 11)(8 9)(10 12)(13 14)(15 16)(17 23)(18 20)(19 21)(22 24)
(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)(13 15)(14 16)(17 21)(18 22)(19 23)(20 24)
(1 8 10)(2 11 5)(3 6 12)(4 9 7)(13 21 20)(14 17 22)(15 23 18)(16 19 24)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 13)(2 16)(3 15)(4 14)(5 19)(6 18)(7 17)(8 20)(9 22)(10 21)(11 24)(12 23)
G:=sub<Sym(24)| (1,13)(2,14)(3,15)(4,16)(5,22)(6,23)(7,24)(8,21)(9,19)(10,20)(11,17)(12,18), (1,2)(3,4)(5,7)(6,11)(8,9)(10,12)(13,14)(15,16)(17,23)(18,20)(19,21)(22,24), (1,3)(2,4)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,21)(18,22)(19,23)(20,24), (1,8,10)(2,11,5)(3,6,12)(4,9,7)(13,21,20)(14,17,22)(15,23,18)(16,19,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,13)(2,16)(3,15)(4,14)(5,19)(6,18)(7,17)(8,20)(9,22)(10,21)(11,24)(12,23)>;
G:=Group( (1,13)(2,14)(3,15)(4,16)(5,22)(6,23)(7,24)(8,21)(9,19)(10,20)(11,17)(12,18), (1,2)(3,4)(5,7)(6,11)(8,9)(10,12)(13,14)(15,16)(17,23)(18,20)(19,21)(22,24), (1,3)(2,4)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,21)(18,22)(19,23)(20,24), (1,8,10)(2,11,5)(3,6,12)(4,9,7)(13,21,20)(14,17,22)(15,23,18)(16,19,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,13)(2,16)(3,15)(4,14)(5,19)(6,18)(7,17)(8,20)(9,22)(10,21)(11,24)(12,23) );
G=PermutationGroup([[(1,13),(2,14),(3,15),(4,16),(5,22),(6,23),(7,24),(8,21),(9,19),(10,20),(11,17),(12,18)], [(1,2),(3,4),(5,7),(6,11),(8,9),(10,12),(13,14),(15,16),(17,23),(18,20),(19,21),(22,24)], [(1,3),(2,4),(5,12),(6,9),(7,10),(8,11),(13,15),(14,16),(17,21),(18,22),(19,23),(20,24)], [(1,8,10),(2,11,5),(3,6,12),(4,9,7),(13,21,20),(14,17,22),(15,23,18),(16,19,24)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,13),(2,16),(3,15),(4,14),(5,19),(6,18),(7,17),(8,20),(9,22),(10,21),(11,24),(12,23)]])
G:=TransitiveGroup(24,399);
►On 24 points - transitive group 24T405
Generators in S24
(1 11)(2 12)(3 9)(4 10)(5 16)(6 13)(7 14)(8 15)(17 21)(18 22)(19 23)(20 24)
(1 3)(2 12)(4 10)(5 14)(6 8)(7 16)(9 11)(13 15)(17 21)(18 24)(19 23)(20 22)
(1 9)(2 10)(3 11)(4 12)(5 16)(6 13)(7 14)(8 15)(17 19)(18 20)(21 23)(22 24)
(1 19 7)(2 8 20)(3 17 5)(4 6 18)(9 21 16)(10 13 22)(11 23 14)(12 15 24)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 4)(2 3)(5 8)(6 7)(9 12)(10 11)(13 14)(15 16)(17 20)(18 19)(21 24)(22 23)
G:=sub<Sym(24)| (1,11)(2,12)(3,9)(4,10)(5,16)(6,13)(7,14)(8,15)(17,21)(18,22)(19,23)(20,24), (1,3)(2,12)(4,10)(5,14)(6,8)(7,16)(9,11)(13,15)(17,21)(18,24)(19,23)(20,22), (1,9)(2,10)(3,11)(4,12)(5,16)(6,13)(7,14)(8,15)(17,19)(18,20)(21,23)(22,24), (1,19,7)(2,8,20)(3,17,5)(4,6,18)(9,21,16)(10,13,22)(11,23,14)(12,15,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,24)(22,23)>;
G:=Group( (1,11)(2,12)(3,9)(4,10)(5,16)(6,13)(7,14)(8,15)(17,21)(18,22)(19,23)(20,24), (1,3)(2,12)(4,10)(5,14)(6,8)(7,16)(9,11)(13,15)(17,21)(18,24)(19,23)(20,22), (1,9)(2,10)(3,11)(4,12)(5,16)(6,13)(7,14)(8,15)(17,19)(18,20)(21,23)(22,24), (1,19,7)(2,8,20)(3,17,5)(4,6,18)(9,21,16)(10,13,22)(11,23,14)(12,15,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,24)(22,23) );
G=PermutationGroup([[(1,11),(2,12),(3,9),(4,10),(5,16),(6,13),(7,14),(8,15),(17,21),(18,22),(19,23),(20,24)], [(1,3),(2,12),(4,10),(5,14),(6,8),(7,16),(9,11),(13,15),(17,21),(18,24),(19,23),(20,22)], [(1,9),(2,10),(3,11),(4,12),(5,16),(6,13),(7,14),(8,15),(17,19),(18,20),(21,23),(22,24)], [(1,19,7),(2,8,20),(3,17,5),(4,6,18),(9,21,16),(10,13,22),(11,23,14),(12,15,24)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11),(13,14),(15,16),(17,20),(18,19),(21,24),(22,23)]])
G:=TransitiveGroup(24,405);
►On 24 points - transitive group 24T406
Generators in S24
(1 10)(2 11)(3 12)(4 9)(5 15)(6 16)(7 13)(8 14)(17 23)(18 24)(19 21)(20 22)
(2 4)(5 7)(6 8)(9 11)(13 15)(14 16)(17 19)(21 23)
(1 3)(2 4)(9 11)(10 12)(17 19)(18 20)(21 23)(22 24)
(1 15 19)(2 20 16)(3 13 17)(4 18 14)(5 21 10)(6 11 22)(7 23 12)(8 9 24)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 9)(2 12)(3 11)(4 10)(5 14)(6 13)(7 16)(8 15)(17 22)(18 21)(19 24)(20 23)
G:=sub<Sym(24)| (1,10)(2,11)(3,12)(4,9)(5,15)(6,16)(7,13)(8,14)(17,23)(18,24)(19,21)(20,22), (2,4)(5,7)(6,8)(9,11)(13,15)(14,16)(17,19)(21,23), (1,3)(2,4)(9,11)(10,12)(17,19)(18,20)(21,23)(22,24), (1,15,19)(2,20,16)(3,13,17)(4,18,14)(5,21,10)(6,11,22)(7,23,12)(8,9,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,9)(2,12)(3,11)(4,10)(5,14)(6,13)(7,16)(8,15)(17,22)(18,21)(19,24)(20,23)>;
G:=Group( (1,10)(2,11)(3,12)(4,9)(5,15)(6,16)(7,13)(8,14)(17,23)(18,24)(19,21)(20,22), (2,4)(5,7)(6,8)(9,11)(13,15)(14,16)(17,19)(21,23), (1,3)(2,4)(9,11)(10,12)(17,19)(18,20)(21,23)(22,24), (1,15,19)(2,20,16)(3,13,17)(4,18,14)(5,21,10)(6,11,22)(7,23,12)(8,9,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,9)(2,12)(3,11)(4,10)(5,14)(6,13)(7,16)(8,15)(17,22)(18,21)(19,24)(20,23) );
G=PermutationGroup([[(1,10),(2,11),(3,12),(4,9),(5,15),(6,16),(7,13),(8,14),(17,23),(18,24),(19,21),(20,22)], [(2,4),(5,7),(6,8),(9,11),(13,15),(14,16),(17,19),(21,23)], [(1,3),(2,4),(9,11),(10,12),(17,19),(18,20),(21,23),(22,24)], [(1,15,19),(2,20,16),(3,13,17),(4,18,14),(5,21,10),(6,11,22),(7,23,12),(8,9,24)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,9),(2,12),(3,11),(4,10),(5,14),(6,13),(7,16),(8,15),(17,22),(18,21),(19,24),(20,23)]])
G:=TransitiveGroup(24,406);
►On 24 points - transitive group 24T407
Generators in S24
(1 6)(2 7)(3 8)(4 5)(9 14)(10 15)(11 16)(12 13)(17 22)(18 23)(19 24)(20 21)
(2 7)(4 5)(9 14)(10 15)(11 16)(12 13)(18 23)(20 21)
(1 6)(2 7)(3 8)(4 5)(17 22)(18 23)(19 24)(20 21)
(1 9 23)(2 24 10)(3 11 21)(4 22 12)(5 17 13)(6 14 18)(7 19 15)(8 16 20)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 4)(2 3)(5 6)(7 8)(9 12)(10 11)(13 14)(15 16)(17 18)(19 20)(21 24)(22 23)
G:=sub<Sym(24)| (1,6)(2,7)(3,8)(4,5)(9,14)(10,15)(11,16)(12,13)(17,22)(18,23)(19,24)(20,21), (2,7)(4,5)(9,14)(10,15)(11,16)(12,13)(18,23)(20,21), (1,6)(2,7)(3,8)(4,5)(17,22)(18,23)(19,24)(20,21), (1,9,23)(2,24,10)(3,11,21)(4,22,12)(5,17,13)(6,14,18)(7,19,15)(8,16,20), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,18)(19,20)(21,24)(22,23)>;
G:=Group( (1,6)(2,7)(3,8)(4,5)(9,14)(10,15)(11,16)(12,13)(17,22)(18,23)(19,24)(20,21), (2,7)(4,5)(9,14)(10,15)(11,16)(12,13)(18,23)(20,21), (1,6)(2,7)(3,8)(4,5)(17,22)(18,23)(19,24)(20,21), (1,9,23)(2,24,10)(3,11,21)(4,22,12)(5,17,13)(6,14,18)(7,19,15)(8,16,20), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,18)(19,20)(21,24)(22,23) );
G=PermutationGroup([[(1,6),(2,7),(3,8),(4,5),(9,14),(10,15),(11,16),(12,13),(17,22),(18,23),(19,24),(20,21)], [(2,7),(4,5),(9,14),(10,15),(11,16),(12,13),(18,23),(20,21)], [(1,6),(2,7),(3,8),(4,5),(17,22),(18,23),(19,24),(20,21)], [(1,9,23),(2,24,10),(3,11,21),(4,22,12),(5,17,13),(6,14,18),(7,19,15),(8,16,20)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11),(13,14),(15,16),(17,18),(19,20),(21,24),(22,23)]])
G:=TransitiveGroup(24,407);
►On 24 points - transitive group 24T416
Generators in S24
(1 9)(2 10)(3 11)(4 12)(5 17)(6 18)(7 19)(8 20)(13 21)(14 22)(15 23)(16 24)
(1 3)(2 4)(6 8)(9 11)(10 12)(14 16)(18 20)(22 24)
(5 7)(6 8)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 20 23)(2 24 17)(3 18 21)(4 22 19)(5 10 16)(6 13 11)(7 12 14)(8 15 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 9)(2 12)(3 11)(4 10)(5 22)(6 21)(7 24)(8 23)(13 18)(14 17)(15 20)(16 19)
G:=sub<Sym(24)| (1,9)(2,10)(3,11)(4,12)(5,17)(6,18)(7,19)(8,20)(13,21)(14,22)(15,23)(16,24), (1,3)(2,4)(6,8)(9,11)(10,12)(14,16)(18,20)(22,24), (5,7)(6,8)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,20,23)(2,24,17)(3,18,21)(4,22,19)(5,10,16)(6,13,11)(7,12,14)(8,15,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,9)(2,12)(3,11)(4,10)(5,22)(6,21)(7,24)(8,23)(13,18)(14,17)(15,20)(16,19)>;
G:=Group( (1,9)(2,10)(3,11)(4,12)(5,17)(6,18)(7,19)(8,20)(13,21)(14,22)(15,23)(16,24), (1,3)(2,4)(6,8)(9,11)(10,12)(14,16)(18,20)(22,24), (5,7)(6,8)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,20,23)(2,24,17)(3,18,21)(4,22,19)(5,10,16)(6,13,11)(7,12,14)(8,15,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,9)(2,12)(3,11)(4,10)(5,22)(6,21)(7,24)(8,23)(13,18)(14,17)(15,20)(16,19) );
G=PermutationGroup([[(1,9),(2,10),(3,11),(4,12),(5,17),(6,18),(7,19),(8,20),(13,21),(14,22),(15,23),(16,24)], [(1,3),(2,4),(6,8),(9,11),(10,12),(14,16),(18,20),(22,24)], [(5,7),(6,8),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)], [(1,20,23),(2,24,17),(3,18,21),(4,22,19),(5,10,16),(6,13,11),(7,12,14),(8,15,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,9),(2,12),(3,11),(4,10),(5,22),(6,21),(7,24),(8,23),(13,18),(14,17),(15,20),(16,19)]])
G:=TransitiveGroup(24,416);
Matrix representation of C2×A4⋊D4 ►in GL7(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 2 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 11 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(7,GF(13))| [12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,9,0,0,0,0,0,0,6,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0],[1,12,0,0,0,0,0,2,12,0,0,0,0,0,0,0,12,1,0,0,0,0,0,11,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,12,0],[1,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0] >;
C2×A4⋊D4 in GAP, Magma, Sage, TeX
C_2\times A_4\rtimes D_4
% in TeX
G:=Group("C2xA4:D4"); // GroupNames label
G:=SmallGroup(192,1488);
// by ID
G=gap.SmallGroup(192,1488);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,254,1124,4037,285,2358,475]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^3=e^4=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,d*b*d^-1=e*b*e^-1=f*b*f=b*c=c*b,d*c*d^-1=b,c*e=e*c,c*f=f*c,e*d*e^-1=f*d*f=d^-1,f*e*f=e^-1>;
// generators/relations
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