metabelian, supersoluble, monomial
Aliases: C62.32D4, C23.9S32, C6.7(D6⋊C4), (C2×C6).57D12, C6.D4⋊2S3, C32⋊5(C23⋊C4), C62.35(C2×C4), (C22×C6).55D6, (C2×C62).8C22, C3⋊1(C23.6D6), C2.8(C6.D12), C22.3(C3⋊D12), C22.4(C6.D6), (C22×C3⋊S3)⋊1C4, (C2×C6).12(C4×S3), (C2×C3⋊Dic3)⋊1C4, (C3×C6.D4)⋊2C2, (C2×C6).13(C3⋊D4), (C2×C32⋊7D4).1C2, (C3×C6).40(C22⋊C4), SmallGroup(288,229)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.32D4
G = < a,b,c,d | a6=b6=c4=1, d2=b3, ab=ba, cac-1=ab3, dad-1=a-1b3, cbc-1=dbd-1=b-1, dcd-1=a3b3c-1 >
Subgroups: 642 in 131 conjugacy classes, 32 normal (12 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C2×D4, C3⋊S3, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C22×C6, C23⋊C4, C3×Dic3, C3⋊Dic3, C2×C3⋊S3, C62, C62, C62, C6.D4, C3×C22⋊C4, C2×C3⋊D4, C6×Dic3, C2×C3⋊Dic3, C32⋊7D4, C22×C3⋊S3, C2×C62, C23.6D6, C3×C6.D4, C2×C32⋊7D4, C62.32D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C4×S3, D12, C3⋊D4, C23⋊C4, S32, D6⋊C4, C6.D6, C3⋊D12, C23.6D6, C6.D12, C62.32D4
►On 24 points - transitive group 24T583
(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 11 9 10 8 12)(13 14 15 16 17 18)(19 24 23 22 21 20)
(1 14)(2 18)(3 16)(4 17)(5 15)(6 13)(7 22 10 19)(8 20 11 23)(9 24 12 21)
(1 20 4 23)(2 22 5 19)(3 24 6 21)(7 15 10 18)(8 17 11 14)(9 13 12 16)
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)(7,11,9,10,8,12)(13,14,15,16,17,18)(19,24,23,22,21,20), (1,14)(2,18)(3,16)(4,17)(5,15)(6,13)(7,22,10,19)(8,20,11,23)(9,24,12,21), (1,20,4,23)(2,22,5,19)(3,24,6,21)(7,15,10,18)(8,17,11,14)(9,13,12,16)>;
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)(7,11,9,10,8,12)(13,14,15,16,17,18)(19,24,23,22,21,20), (1,14)(2,18)(3,16)(4,17)(5,15)(6,13)(7,22,10,19)(8,20,11,23)(9,24,12,21), (1,20,4,23)(2,22,5,19)(3,24,6,21)(7,15,10,18)(8,17,11,14)(9,13,12,16) );
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),(7,11,9,10,8,12),(13,14,15,16,17,18),(19,24,23,22,21,20)], [(1,14),(2,18),(3,16),(4,17),(5,15),(6,13),(7,22,10,19),(8,20,11,23),(9,24,12,21)], [(1,20,4,23),(2,22,5,19),(3,24,6,21),(7,15,10,18),(8,17,11,14),(9,13,12,16)]])
G:=TransitiveGroup(24,583);
►On 24 points - transitive group 24T615
(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 16 3 18 5 14)(2 17 4 13 6 15)(7 20 11 24 9 22)(8 21 12 19 10 23)
(1 21 4 7)(2 11 5 19)(3 23 6 9)(8 13 22 16)(10 15 24 18)(12 17 20 14)
(1 4 18 15)(2 14 13 3)(5 6 16 17)(7 21 24 10)(8 9 19 20)(11 23 22 12)
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), (1,16,3,18,5,14)(2,17,4,13,6,15)(7,20,11,24,9,22)(8,21,12,19,10,23), (1,21,4,7)(2,11,5,19)(3,23,6,9)(8,13,22,16)(10,15,24,18)(12,17,20,14), (1,4,18,15)(2,14,13,3)(5,6,16,17)(7,21,24,10)(8,9,19,20)(11,23,22,12)>;
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), (1,16,3,18,5,14)(2,17,4,13,6,15)(7,20,11,24,9,22)(8,21,12,19,10,23), (1,21,4,7)(2,11,5,19)(3,23,6,9)(8,13,22,16)(10,15,24,18)(12,17,20,14), (1,4,18,15)(2,14,13,3)(5,6,16,17)(7,21,24,10)(8,9,19,20)(11,23,22,12) );
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)], [(1,16,3,18,5,14),(2,17,4,13,6,15),(7,20,11,24,9,22),(8,21,12,19,10,23)], [(1,21,4,7),(2,11,5,19),(3,23,6,9),(8,13,22,16),(10,15,24,18),(12,17,20,14)], [(1,4,18,15),(2,14,13,3),(5,6,16,17),(7,21,24,10),(8,9,19,20),(11,23,22,12)]])
G:=TransitiveGroup(24,615);
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 6A | ··· | 6F | 6G | ··· | 6Q | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 2 | 2 | 36 | 2 | 2 | 4 | 12 | 12 | 12 | 12 | 36 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | ··· | 12 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | D4 | D6 | C4×S3 | D12 | C3⋊D4 | C23⋊C4 | S32 | C6.D6 | C3⋊D12 | C23.6D6 | C62.32D4 |
| kernel | C62.32D4 | C3×C6.D4 | C2×C32⋊7D4 | C2×C3⋊Dic3 | C22×C3⋊S3 | C6.D4 | C62 | C22×C6 | C2×C6 | C2×C6 | C2×C6 | C32 | C23 | C22 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 2 | 4 | 4 |
Matrix representation of C62.32D4 ►in GL4(𝔽7) generated by
| 5 | 2 | 3 | 2 |
| 1 | 4 | 3 | 4 |
| 2 | 2 | 3 | 3 |
| 0 | 0 | 0 | 2 |
| 0 | 2 | 5 | 1 |
| 0 | 5 | 0 | 5 |
| 4 | 4 | 1 | 6 |
| 0 | 0 | 0 | 3 |
| 0 | 3 | 6 | 1 |
| 1 | 1 | 4 | 4 |
| 4 | 3 | 5 | 3 |
| 3 | 3 | 2 | 1 |
| 4 | 5 | 2 | 6 |
| 5 | 6 | 0 | 2 |
| 5 | 5 | 4 | 4 |
| 3 | 4 | 2 | 0 |
G:=sub<GL(4,GF(7))| [5,1,2,0,2,4,2,0,3,3,3,0,2,4,3,2],[0,0,4,0,2,5,4,0,5,0,1,0,1,5,6,3],[0,1,4,3,3,1,3,3,6,4,5,2,1,4,3,1],[4,5,5,3,5,6,5,4,2,0,4,2,6,2,4,0] >;
C62.32D4 in GAP, Magma, Sage, TeX
C_6^2._{32}D_4 % in TeX
G:=Group("C6^2.32D4"); // GroupNames label
G:=SmallGroup(288,229);
// by ID
G=gap.SmallGroup(288,229);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,92,219,675,346,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^4=1,d^2=b^3,a*b=b*a,c*a*c^-1=a*b^3,d*a*d^-1=a^-1*b^3,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^3*b^3*c^-1>;
// generators/relations