non-abelian, soluble, monomial
Aliases: C62.2D4, C6.D6⋊C4, C32⋊(C23⋊C4), C22.2S3≀C2, C62⋊C4⋊1C2, Dic3⋊D6.1C2, C6.D12⋊1C2, (C2×S32)⋊C4, C2.8(S32⋊C4), (C2×C3⋊S3).9D4, (C3×C6).8(C22⋊C4), (C22×C3⋊S3).2C22, (C2×C3⋊S3).10(C2×C4), SmallGroup(288,386)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.2D4
G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=a3b, dad=a-1b3, cbc-1=a2b3, bd=db, dcd=a3c-1 >
Subgroups: 640 in 96 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, D4, C23, C32, Dic3, C12, D6, C2×C6, C22⋊C4, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C23⋊C4, C3×Dic3, C32⋊C4, S32, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, D6⋊C4, S3×D4, C6.D6, C3⋊D12, C6×Dic3, C3×C3⋊D4, C2×C32⋊C4, C2×S32, C22×C3⋊S3, C6.D12, C62⋊C4, Dic3⋊D6, C62.2D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, C23⋊C4, S3≀C2, S32⋊C4, C62.2D4
Character table of C62.2D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 12C | 12D | 12E | |
| size | 1 | 1 | 2 | 12 | 18 | 18 | 4 | 4 | 12 | 12 | 12 | 36 | 36 | 4 | 4 | 4 | 4 | 8 | 24 | 12 | 12 | 12 | 12 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | i | -i | 1 | i | -i | 1 | 1 | -1 | -1 | -1 | -1 | -i | i | i | -i | 1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | i | -i | -1 | -i | i | 1 | 1 | -1 | -1 | -1 | 1 | -i | i | i | -i | -1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -i | i | -1 | i | -i | 1 | 1 | -1 | -1 | -1 | 1 | i | -i | -i | i | -1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -i | i | 1 | -i | i | 1 | 1 | -1 | -1 | -1 | -1 | i | -i | -i | i | 1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | 0 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 4 | 4 | -4 | -2 | 0 | 0 | 1 | -2 | 0 | 0 | 2 | 0 | 0 | 1 | -2 | 2 | 2 | -1 | 1 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from S32⋊C4 |
| ρ12 | 4 | 4 | -4 | 2 | 0 | 0 | 1 | -2 | 0 | 0 | -2 | 0 | 0 | 1 | -2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | orthogonal lifted from S32⋊C4 |
| ρ13 | 4 | 4 | 4 | 2 | 0 | 0 | 1 | -2 | 0 | 0 | 2 | 0 | 0 | 1 | -2 | -2 | -2 | 1 | -1 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from S3≀C2 |
| ρ14 | 4 | 4 | 4 | -2 | 0 | 0 | 1 | -2 | 0 | 0 | -2 | 0 | 0 | 1 | -2 | -2 | -2 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | orthogonal lifted from S3≀C2 |
| ρ15 | 4 | 4 | 4 | 0 | 0 | 0 | -2 | 1 | 2 | 2 | 0 | 0 | 0 | -2 | 1 | 1 | 1 | -2 | 0 | -1 | -1 | -1 | -1 | 0 | orthogonal lifted from S3≀C2 |
| ρ16 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ17 | 4 | 4 | 4 | 0 | 0 | 0 | -2 | 1 | -2 | -2 | 0 | 0 | 0 | -2 | 1 | 1 | 1 | -2 | 0 | 1 | 1 | 1 | 1 | 0 | orthogonal lifted from S3≀C2 |
| ρ18 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | 3 | -3 | 0 | 0 | √3 | -√3 | √3 | -√3 | 0 | orthogonal faithful |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | 3 | -3 | 0 | 0 | -√3 | √3 | -√3 | √3 | 0 | orthogonal faithful |
| ρ20 | 4 | 4 | -4 | 0 | 0 | 0 | -2 | 1 | -2i | 2i | 0 | 0 | 0 | -2 | 1 | -1 | -1 | 2 | 0 | -i | i | i | -i | 0 | complex lifted from S32⋊C4 |
| ρ21 | 4 | 4 | -4 | 0 | 0 | 0 | -2 | 1 | 2i | -2i | 0 | 0 | 0 | -2 | 1 | -1 | -1 | 2 | 0 | i | -i | -i | i | 0 | complex lifted from S32⋊C4 |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | -3 | 3 | 0 | 0 | -√-3 | -√-3 | √-3 | √-3 | 0 | complex faithful |
| ρ23 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | -3 | 3 | 0 | 0 | √-3 | √-3 | -√-3 | -√-3 | 0 | complex faithful |
| ρ24 | 8 | -8 | 0 | 0 | 0 | 0 | 2 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 24 points - transitive group 24T596
Generators in S24
(7 8)(9 10)(11 12)(13 14 15)(16 17 18)(19 20 21 22 23 24)
(1 5 3 4 2 6)(7 11 9 8 12 10)(13 18)(14 16)(15 17)(19 22)(20 23)(21 24)
(1 22 4 19)(2 20 6 21)(3 24 5 23)(7 13 12 14)(8 18 11 16)(9 15)(10 17)
(1 10)(2 8)(3 11)(4 9)(5 7)(6 12)(13 24)(14 20)(15 22)(16 23)(17 19)(18 21)
G:=sub<Sym(24)| (7,8)(9,10)(11,12)(13,14,15)(16,17,18)(19,20,21,22,23,24), (1,5,3,4,2,6)(7,11,9,8,12,10)(13,18)(14,16)(15,17)(19,22)(20,23)(21,24), (1,22,4,19)(2,20,6,21)(3,24,5,23)(7,13,12,14)(8,18,11,16)(9,15)(10,17), (1,10)(2,8)(3,11)(4,9)(5,7)(6,12)(13,24)(14,20)(15,22)(16,23)(17,19)(18,21)>;
G:=Group( (7,8)(9,10)(11,12)(13,14,15)(16,17,18)(19,20,21,22,23,24), (1,5,3,4,2,6)(7,11,9,8,12,10)(13,18)(14,16)(15,17)(19,22)(20,23)(21,24), (1,22,4,19)(2,20,6,21)(3,24,5,23)(7,13,12,14)(8,18,11,16)(9,15)(10,17), (1,10)(2,8)(3,11)(4,9)(5,7)(6,12)(13,24)(14,20)(15,22)(16,23)(17,19)(18,21) );
G=PermutationGroup([[(7,8),(9,10),(11,12),(13,14,15),(16,17,18),(19,20,21,22,23,24)], [(1,5,3,4,2,6),(7,11,9,8,12,10),(13,18),(14,16),(15,17),(19,22),(20,23),(21,24)], [(1,22,4,19),(2,20,6,21),(3,24,5,23),(7,13,12,14),(8,18,11,16),(9,15),(10,17)], [(1,10),(2,8),(3,11),(4,9),(5,7),(6,12),(13,24),(14,20),(15,22),(16,23),(17,19),(18,21)]])
G:=TransitiveGroup(24,596);
►On 24 points - transitive group 24T597
Generators in S24
(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 5 3 4 2 6)(7 11 9 10 8 12)(13 18 17 16 15 14)(19 24 23 22 21 20)
(1 20 4 23)(2 24 6 19)(3 22 5 21)(7 17)(8 15 9 13)(10 14)(11 18 12 16)
(1 17)(2 13)(3 15)(4 14)(5 16)(6 18)(7 23)(8 19)(9 21)(10 20)(11 22)(12 24)
G:=sub<Sym(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), (1,5,3,4,2,6)(7,11,9,10,8,12)(13,18,17,16,15,14)(19,24,23,22,21,20), (1,20,4,23)(2,24,6,19)(3,22,5,21)(7,17)(8,15,9,13)(10,14)(11,18,12,16), (1,17)(2,13)(3,15)(4,14)(5,16)(6,18)(7,23)(8,19)(9,21)(10,20)(11,22)(12,24)>;
G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,5,3,4,2,6)(7,11,9,10,8,12)(13,18,17,16,15,14)(19,24,23,22,21,20), (1,20,4,23)(2,24,6,19)(3,22,5,21)(7,17)(8,15,9,13)(10,14)(11,18,12,16), (1,17)(2,13)(3,15)(4,14)(5,16)(6,18)(7,23)(8,19)(9,21)(10,20)(11,22)(12,24) );
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,5,3,4,2,6),(7,11,9,10,8,12),(13,18,17,16,15,14),(19,24,23,22,21,20)], [(1,20,4,23),(2,24,6,19),(3,22,5,21),(7,17),(8,15,9,13),(10,14),(11,18,12,16)], [(1,17),(2,13),(3,15),(4,14),(5,16),(6,18),(7,23),(8,19),(9,21),(10,20),(11,22),(12,24)]])
G:=TransitiveGroup(24,597);
►On 24 points - transitive group 24T598
Generators in S24
(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 6 2 4 3 5)(7 12 8 10 9 11)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 24 6 19)(2 20 5 23)(3 22 4 21)(7 13)(8 15 9 17)(10 16)(11 18 12 14)
(1 17)(2 13)(3 15)(4 14)(5 16)(6 18)(7 23)(8 19)(9 21)(10 20)(11 22)(12 24)
G:=sub<Sym(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), (1,6,2,4,3,5)(7,12,8,10,9,11)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,24,6,19)(2,20,5,23)(3,22,4,21)(7,13)(8,15,9,17)(10,16)(11,18,12,14), (1,17)(2,13)(3,15)(4,14)(5,16)(6,18)(7,23)(8,19)(9,21)(10,20)(11,22)(12,24)>;
G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,6,2,4,3,5)(7,12,8,10,9,11)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,24,6,19)(2,20,5,23)(3,22,4,21)(7,13)(8,15,9,17)(10,16)(11,18,12,14), (1,17)(2,13)(3,15)(4,14)(5,16)(6,18)(7,23)(8,19)(9,21)(10,20)(11,22)(12,24) );
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,6,2,4,3,5),(7,12,8,10,9,11),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,24,6,19),(2,20,5,23),(3,22,4,21),(7,13),(8,15,9,17),(10,16),(11,18,12,14)], [(1,17),(2,13),(3,15),(4,14),(5,16),(6,18),(7,23),(8,19),(9,21),(10,20),(11,22),(12,24)]])
G:=TransitiveGroup(24,598);
►On 24 points - transitive group 24T602
Generators in S24
(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 5 3 4 2 6)(7 11 9 10 8 12)(13 18 17 16 15 14)(19 24 23 22 21 20)
(1 22 11 13)(2 20 10 15)(3 24 12 17)(4 19 8 16)(5 23 7 18)(6 21 9 14)
(1 13)(2 15)(3 17)(4 16)(5 18)(6 14)(7 20)(8 22)(9 24)(10 23)(11 19)(12 21)
G:=sub<Sym(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), (1,5,3,4,2,6)(7,11,9,10,8,12)(13,18,17,16,15,14)(19,24,23,22,21,20), (1,22,11,13)(2,20,10,15)(3,24,12,17)(4,19,8,16)(5,23,7,18)(6,21,9,14), (1,13)(2,15)(3,17)(4,16)(5,18)(6,14)(7,20)(8,22)(9,24)(10,23)(11,19)(12,21)>;
G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,5,3,4,2,6)(7,11,9,10,8,12)(13,18,17,16,15,14)(19,24,23,22,21,20), (1,22,11,13)(2,20,10,15)(3,24,12,17)(4,19,8,16)(5,23,7,18)(6,21,9,14), (1,13)(2,15)(3,17)(4,16)(5,18)(6,14)(7,20)(8,22)(9,24)(10,23)(11,19)(12,21) );
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,5,3,4,2,6),(7,11,9,10,8,12),(13,18,17,16,15,14),(19,24,23,22,21,20)], [(1,22,11,13),(2,20,10,15),(3,24,12,17),(4,19,8,16),(5,23,7,18),(6,21,9,14)], [(1,13),(2,15),(3,17),(4,16),(5,18),(6,14),(7,20),(8,22),(9,24),(10,23),(11,19),(12,21)]])
G:=TransitiveGroup(24,602);
►On 24 points - transitive group 24T605
Generators in S24
(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 5 3 6)(7 12 8 10 9 11)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 23 10 16)(2 19 12 14)(3 21 11 18)(4 24 8 15)(5 20 7 13)(6 22 9 17)
(1 16)(2 18)(3 14)(4 17)(5 13)(6 15)(7 23)(8 19)(9 21)(10 20)(11 22)(12 24)
G:=sub<Sym(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), (1,4,2,5,3,6)(7,12,8,10,9,11)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,23,10,16)(2,19,12,14)(3,21,11,18)(4,24,8,15)(5,20,7,13)(6,22,9,17), (1,16)(2,18)(3,14)(4,17)(5,13)(6,15)(7,23)(8,19)(9,21)(10,20)(11,22)(12,24)>;
G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,4,2,5,3,6)(7,12,8,10,9,11)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,23,10,16)(2,19,12,14)(3,21,11,18)(4,24,8,15)(5,20,7,13)(6,22,9,17), (1,16)(2,18)(3,14)(4,17)(5,13)(6,15)(7,23)(8,19)(9,21)(10,20)(11,22)(12,24) );
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,4,2,5,3,6),(7,12,8,10,9,11),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,23,10,16),(2,19,12,14),(3,21,11,18),(4,24,8,15),(5,20,7,13),(6,22,9,17)], [(1,16),(2,18),(3,14),(4,17),(5,13),(6,15),(7,23),(8,19),(9,21),(10,20),(11,22),(12,24)]])
G:=TransitiveGroup(24,605);
►On 24 points - transitive group 24T665
Generators in S24
(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 5 4 9 11 7)(2 6 3 10 12 8)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 23 6 24)(2 14 5 15)(3 16 7 13)(4 19 8 22)(9 17 12 18)(10 20 11 21)
(1 9)(4 7)(5 11)(14 24)(15 17)(16 22)(18 20)(21 23)
G:=sub<Sym(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), (1,5,4,9,11,7)(2,6,3,10,12,8)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,23,6,24)(2,14,5,15)(3,16,7,13)(4,19,8,22)(9,17,12,18)(10,20,11,21), (1,9)(4,7)(5,11)(14,24)(15,17)(16,22)(18,20)(21,23)>;
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,5,4,9,11,7)(2,6,3,10,12,8)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,23,6,24)(2,14,5,15)(3,16,7,13)(4,19,8,22)(9,17,12,18)(10,20,11,21), (1,9)(4,7)(5,11)(14,24)(15,17)(16,22)(18,20)(21,23) );
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,5,4,9,11,7),(2,6,3,10,12,8),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,23,6,24),(2,14,5,15),(3,16,7,13),(4,19,8,22),(9,17,12,18),(10,20,11,21)], [(1,9),(4,7),(5,11),(14,24),(15,17),(16,22),(18,20),(21,23)]])
G:=TransitiveGroup(24,665);
Matrix representation of C62.2D4 ►in GL4(𝔽7) generated by
| 6 | 4 | 6 | 1 |
| 1 | 3 | 0 | 5 |
| 4 | 5 | 6 | 3 |
| 3 | 5 | 0 | 6 |
| 1 | 0 | 4 | 6 |
| 2 | 4 | 1 | 0 |
| 6 | 1 | 3 | 2 |
| 4 | 4 | 5 | 1 |
| 5 | 1 | 4 | 1 |
| 6 | 1 | 2 | 1 |
| 5 | 0 | 3 | 0 |
| 1 | 6 | 3 | 5 |
| 1 | 1 | 1 | 0 |
| 2 | 1 | 5 | 4 |
| 5 | 5 | 1 | 3 |
| 5 | 2 | 4 | 4 |
G:=sub<GL(4,GF(7))| [6,1,4,3,4,3,5,5,6,0,6,0,1,5,3,6],[1,2,6,4,0,4,1,4,4,1,3,5,6,0,2,1],[5,6,5,1,1,1,0,6,4,2,3,3,1,1,0,5],[1,2,5,5,1,1,5,2,1,5,1,4,0,4,3,4] >;
C62.2D4 in GAP, Magma, Sage, TeX
C_6^2._2D_4
% in TeX
G:=Group("C6^2.2D4"); // GroupNames label
G:=SmallGroup(288,386);
// by ID
G=gap.SmallGroup(288,386);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,85,64,219,675,2693,2028,691,797,2372]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^3*b,d*a*d=a^-1*b^3,c*b*c^-1=a^2*b^3,b*d=d*b,d*c*d=a^3*c^-1>;
// generators/relations
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