metabelian, supersoluble, monomial
Aliases: Dic6⋊4D6, C3⋊C8⋊15D6, D4.7S32, D4.S3⋊2S3, C3⋊S3⋊4SD16, (C3×D4).9D6, C3⋊3(S3×SD16), C6.57(S3×D4), C3⋊Dic3.20D4, (C3×C12).7C23, C12.7(C22×S3), C32⋊10(C2×SD16), Dic3.D6⋊3C2, C12.29D6⋊4C2, (C3×Dic6)⋊7C22, C32⋊5SD16⋊11C2, C2.17(Dic3⋊D6), C12⋊S3.7C22, (D4×C32).3C22, C4.7(C2×S32), (D4×C3⋊S3).2C2, (C3×C3⋊C8)⋊15C22, (C3×D4.S3)⋊6C2, (C2×C3⋊S3).57D4, (C3×C6).122(C2×D4), (C4×C3⋊S3).13C22, SmallGroup(288,578)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6⋊D6
G = < a,b,c,d | a12=c6=d2=1, b2=a6, bab-1=dad=a-1, cac-1=a7, cbc-1=dbd=a9b, dcd=c-1 >
Subgroups: 834 in 163 conjugacy classes, 40 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, D4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C8, SD16, C2×D4, C2×Q8, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, Dic6, C4×S3, D12, C3⋊D4, C3×D4, C3×D4, C3×Q8, C22×S3, C2×SD16, C3×Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, S3×C8, C24⋊C2, D4.S3, Q8⋊2S3, C3×SD16, S3×D4, S3×Q8, C3×C3⋊C8, C6.D6, C32⋊2Q8, C3×Dic6, C4×C3⋊S3, C12⋊S3, C32⋊7D4, D4×C32, C22×C3⋊S3, S3×SD16, C12.29D6, C32⋊5SD16, C3×D4.S3, Dic3.D6, D4×C3⋊S3, Dic6⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C22×S3, C2×SD16, S32, S3×D4, C2×S32, S3×SD16, Dic3⋊D6, Dic6⋊D6
►On 24 points - transitive group 24T670
(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 7 19)(2 24 8 18)(3 23 9 17)(4 22 10 16)(5 21 11 15)(6 20 12 14)
(1 5 9)(2 12 10 8 6 4)(3 7 11)(13 24 17 16 21 20)(14 19 18 23 22 15)
(1 3)(4 12)(5 11)(6 10)(7 9)(13 14)(15 24)(16 23)(17 22)(18 21)(19 20)
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), (1,13,7,19)(2,24,8,18)(3,23,9,17)(4,22,10,16)(5,21,11,15)(6,20,12,14), (1,5,9)(2,12,10,8,6,4)(3,7,11)(13,24,17,16,21,20)(14,19,18,23,22,15), (1,3)(4,12)(5,11)(6,10)(7,9)(13,14)(15,24)(16,23)(17,22)(18,21)(19,20)>;
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), (1,13,7,19)(2,24,8,18)(3,23,9,17)(4,22,10,16)(5,21,11,15)(6,20,12,14), (1,5,9)(2,12,10,8,6,4)(3,7,11)(13,24,17,16,21,20)(14,19,18,23,22,15), (1,3)(4,12)(5,11)(6,10)(7,9)(13,14)(15,24)(16,23)(17,22)(18,21)(19,20) );
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)], [(1,13,7,19),(2,24,8,18),(3,23,9,17),(4,22,10,16),(5,21,11,15),(6,20,12,14)], [(1,5,9),(2,12,10,8,6,4),(3,7,11),(13,24,17,16,21,20),(14,19,18,23,22,15)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,14),(15,24),(16,23),(17,22),(18,21),(19,20)]])
G:=TransitiveGroup(24,670);
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 4 | 9 | 9 | 36 | 2 | 2 | 4 | 2 | 12 | 12 | 18 | 2 | 2 | 4 | 8 | 8 | 8 | 8 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | 24 | 24 | 12 | 12 | 12 | 12 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | SD16 | S32 | S3×D4 | C2×S32 | S3×SD16 | Dic3⋊D6 | Dic6⋊D6 |
| kernel | Dic6⋊D6 | C12.29D6 | C32⋊5SD16 | C3×D4.S3 | Dic3.D6 | D4×C3⋊S3 | D4.S3 | C3⋊Dic3 | C2×C3⋊S3 | C3⋊C8 | Dic6 | C3×D4 | C3⋊S3 | D4 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 2 | 1 | 4 | 2 | 1 |
Matrix representation of Dic6⋊D6 ►in GL6(𝔽73)
| 72 | 48 | 0 | 0 | 0 | 0 |
| 3 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 12 | 4 | 0 | 0 | 0 | 0 |
| 55 | 61 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 70 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 3 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(73))| [72,3,0,0,0,0,48,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[12,55,0,0,0,0,4,61,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,70,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,3,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
Dic6⋊D6 in GAP, Magma, Sage, TeX
{\rm Dic}_6\rtimes D_6 % in TeX
G:=Group("Dic6:D6"); // GroupNames label
G:=SmallGroup(288,578);
// by ID
G=gap.SmallGroup(288,578);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,254,135,675,346,185,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^6=d^2=1,b^2=a^6,b*a*b^-1=d*a*d=a^-1,c*a*c^-1=a^7,c*b*c^-1=d*b*d=a^9*b,d*c*d=c^-1>;
// generators/relations