metabelian, supersoluble, monomial
Aliases: C62⋊24D6, C33⋊28(C2×D4), C3⋊Dic3⋊10D6, C32⋊19(S3×D4), C32⋊7D4⋊8S3, C3⋊5(Dic3⋊D6), C33⋊9D4⋊12C2, (C3×C62)⋊6C22, (C32×C6).74C23, C22⋊2(C32⋊4D6), (C2×C6)⋊7S32, C3⋊5(S3×C3⋊D4), C6.103(C2×S32), (C2×C3⋊S3)⋊23D6, (C3×C3⋊S3)⋊12D4, C33⋊9(C2×C4)⋊6C2, C3⋊S3⋊5(C3⋊D4), (C22×C3⋊S3)⋊12S3, (C6×C3⋊S3)⋊20C22, (C3×C32⋊7D4)⋊6C2, C32⋊15(C2×C3⋊D4), (C2×C32⋊4D6)⋊4C2, (C3×C3⋊Dic3)⋊7C22, (C3×C6).124(C22×S3), C2.10(C2×C32⋊4D6), (C2×C6×C3⋊S3)⋊7C2, SmallGroup(432,696)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62⋊24D6
G = < a,b,c,d | a6=b6=c6=d2=1, ab=ba, cac-1=a-1b3, dad=ab3, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 1640 in 290 conjugacy classes, 51 normal (21 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C32, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C22×S3, C22×C6, C33, C3×Dic3, C3⋊Dic3, S32, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, S3×D4, C2×C3⋊D4, C3×C3⋊S3, C3×C3⋊S3, C32×C6, C32×C6, S3×Dic3, C6.D6, D6⋊S3, C3⋊D12, C3×C3⋊D4, C32⋊7D4, C2×S32, S3×C2×C6, C22×C3⋊S3, C3×C3⋊Dic3, C32⋊4D6, C6×C3⋊S3, C6×C3⋊S3, C6×C3⋊S3, C3×C62, S3×C3⋊D4, Dic3⋊D6, C33⋊9(C2×C4), C33⋊9D4, C3×C32⋊7D4, C2×C32⋊4D6, C2×C6×C3⋊S3, C62⋊24D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, S32, S3×D4, C2×C3⋊D4, C2×S32, C32⋊4D6, S3×C3⋊D4, Dic3⋊D6, C2×C32⋊4D6, C62⋊24D6
►On 24 points - transitive group 24T1284
(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 3 8 2 7)(4 12 5 10 6 11)(13 18 17 16 15 14)(19 20 21 22 23 24)
(1 22 2 24 3 20)(4 17 5 13 6 15)(7 23 8 19 9 21)(10 14 11 16 12 18)
(1 13)(2 17)(3 15)(4 24)(5 22)(6 20)(7 18)(8 16)(9 14)(10 21)(11 19)(12 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), (1,9,3,8,2,7)(4,12,5,10,6,11)(13,18,17,16,15,14)(19,20,21,22,23,24), (1,22,2,24,3,20)(4,17,5,13,6,15)(7,23,8,19,9,21)(10,14,11,16,12,18), (1,13)(2,17)(3,15)(4,24)(5,22)(6,20)(7,18)(8,16)(9,14)(10,21)(11,19)(12,23)>;
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), (1,9,3,8,2,7)(4,12,5,10,6,11)(13,18,17,16,15,14)(19,20,21,22,23,24), (1,22,2,24,3,20)(4,17,5,13,6,15)(7,23,8,19,9,21)(10,14,11,16,12,18), (1,13)(2,17)(3,15)(4,24)(5,22)(6,20)(7,18)(8,16)(9,14)(10,21)(11,19)(12,23) );
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)], [(1,9,3,8,2,7),(4,12,5,10,6,11),(13,18,17,16,15,14),(19,20,21,22,23,24)], [(1,22,2,24,3,20),(4,17,5,13,6,15),(7,23,8,19,9,21),(10,14,11,16,12,18)], [(1,13),(2,17),(3,15),(4,24),(5,22),(6,20),(7,18),(8,16),(9,14),(10,21),(11,19),(12,23)]])
G:=TransitiveGroup(24,1284);
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | ··· | 3H | 4A | 4B | 6A | ··· | 6E | 6F | ··· | 6V | 6W | 6X | 6Y | 6Z | 6AA | 6AB | 12A | 12B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 |
| size | 1 | 1 | 2 | 9 | 9 | 18 | 18 | 18 | 2 | 2 | 2 | 4 | ··· | 4 | 18 | 18 | 2 | ··· | 2 | 4 | ··· | 4 | 18 | 18 | 18 | 18 | 36 | 36 | 36 | 36 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D6 | D6 | D6 | C3⋊D4 | S32 | S3×D4 | C2×S32 | C32⋊4D6 | S3×C3⋊D4 | Dic3⋊D6 | C2×C32⋊4D6 | C62⋊24D6 |
| kernel | C62⋊24D6 | C33⋊9(C2×C4) | C33⋊9D4 | C3×C32⋊7D4 | C2×C32⋊4D6 | C2×C6×C3⋊S3 | C32⋊7D4 | C22×C3⋊S3 | C3×C3⋊S3 | C3⋊Dic3 | C2×C3⋊S3 | C62 | C3⋊S3 | C2×C6 | C32 | C6 | C22 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 3 | 4 | 3 | 2 | 3 | 2 | 4 | 2 | 2 | 4 |
Matrix representation of C62⋊24D6 ►in GL4(𝔽7) generated by
| 2 | 0 | 4 | 2 |
| 5 | 0 | 1 | 6 |
| 4 | 4 | 1 | 6 |
| 0 | 0 | 0 | 4 |
| 4 | 6 | 3 | 2 |
| 6 | 4 | 4 | 2 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 5 |
| 2 | 4 | 2 | 4 |
| 1 | 1 | 0 | 6 |
| 5 | 2 | 1 | 4 |
| 2 | 2 | 6 | 3 |
| 1 | 4 | 5 | 4 |
| 3 | 4 | 4 | 0 |
| 5 | 2 | 1 | 4 |
| 3 | 3 | 4 | 1 |
G:=sub<GL(4,GF(7))| [2,5,4,0,0,0,4,0,4,1,1,0,2,6,6,4],[4,6,0,0,6,4,0,0,3,4,3,0,2,2,0,5],[2,1,5,2,4,1,2,2,2,0,1,6,4,6,4,3],[1,3,5,3,4,4,2,3,5,4,1,4,4,0,4,1] >;
C62⋊24D6 in GAP, Magma, Sage, TeX
C_6^2\rtimes_{24}D_6 % in TeX
G:=Group("C6^2:24D6"); // GroupNames label
G:=SmallGroup(432,696);
// by ID
G=gap.SmallGroup(432,696);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,135,1124,571,2028,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^6=d^2=1,a*b=b*a,c*a*c^-1=a^-1*b^3,d*a*d=a*b^3,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations