direct product, metabelian, supersoluble, monomial
Aliases: C3×Dic3⋊D6, C62⋊20D6, D6⋊3(S3×C6), (S3×C6)⋊13D6, C33⋊20(C2×D4), C3⋊D12⋊6C6, C62⋊10(C2×C6), Dic3⋊2(S3×C6), C6.D6⋊3C6, C32⋊10(C6×D4), C32⋊23(S3×D4), (C3×Dic3)⋊11D6, (C3×C62)⋊2C22, (C32×C6).37C23, (C32×Dic3)⋊6C22, C3⋊3(C3×S3×D4), (C2×S32)⋊4C6, (S32×C6)⋊8C2, (C2×C6)⋊11S32, C2.18(S32×C6), (C2×C6)⋊6(S3×C6), C3⋊S3⋊4(C3×D4), C22⋊4(C3×S32), C6.18(S3×C2×C6), (S3×C6)⋊3(C2×C6), C6.121(C2×S32), (C3×C3⋊S3)⋊10D4, C3⋊D4⋊2(C3×S3), (C3×C3⋊D4)⋊4C6, (C3×C3⋊D4)⋊6S3, (C22×C3⋊S3)⋊9C6, (S3×C3×C6)⋊10C22, (C6×C3⋊S3)⋊15C22, (C3×Dic3)⋊2(C2×C6), (C3×C6.D6)⋊6C2, (C32×C3⋊D4)⋊4C2, (C3×C3⋊D12)⋊12C2, (C3×C6).28(C22×C6), (C3×C6).142(C22×S3), (C2×C6×C3⋊S3)⋊2C2, (C2×C3⋊S3)⋊8(C2×C6), SmallGroup(432,659)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Dic3⋊D6
G = < a,b,c,d,e | a3=b6=d6=e2=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, dcd-1=b3c, ce=ec, ede=d-1 >
Subgroups: 1152 in 290 conjugacy classes, 68 normal (24 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C32, C32, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, D12, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C33, C3×Dic3, C3×Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, S3×D4, C6×D4, S3×C32, C3×C3⋊S3, C3×C3⋊S3, C32×C6, C32×C6, C6.D6, C3⋊D12, S3×C12, C3×D12, C3×C3⋊D4, C3×C3⋊D4, D4×C32, C2×S32, S3×C2×C6, C22×C3⋊S3, C32×Dic3, C3×S32, S3×C3×C6, C6×C3⋊S3, C6×C3⋊S3, C3×C62, Dic3⋊D6, C3×S3×D4, C3×C6.D6, C3×C3⋊D12, C32×C3⋊D4, S32×C6, C2×C6×C3⋊S3, C3×Dic3⋊D6
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, C3×D4, C22×S3, C22×C6, S32, S3×C6, S3×D4, C6×D4, C2×S32, S3×C2×C6, C3×S32, Dic3⋊D6, C3×S3×D4, S32×C6, C3×Dic3⋊D6
►On 24 points - transitive group 24T1282
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 23 21)(20 24 22)
(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 10 4 7)(2 9 5 12)(3 8 6 11)(13 23 16 20)(14 22 17 19)(15 21 18 24)
(1 5 3)(2 6 4)(7 12 11 10 9 8)(13 18 17 16 15 14)(19 23 21)(20 24 22)
(1 21)(2 20)(3 19)(4 24)(5 23)(6 22)(7 15)(8 14)(9 13)(10 18)(11 17)(12 16)
G:=sub<Sym(24)| (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,10,4,7)(2,9,5,12)(3,8,6,11)(13,23,16,20)(14,22,17,19)(15,21,18,24), (1,5,3)(2,6,4)(7,12,11,10,9,8)(13,18,17,16,15,14)(19,23,21)(20,24,22), (1,21)(2,20)(3,19)(4,24)(5,23)(6,22)(7,15)(8,14)(9,13)(10,18)(11,17)(12,16)>;
G:=Group( (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,10,4,7)(2,9,5,12)(3,8,6,11)(13,23,16,20)(14,22,17,19)(15,21,18,24), (1,5,3)(2,6,4)(7,12,11,10,9,8)(13,18,17,16,15,14)(19,23,21)(20,24,22), (1,21)(2,20)(3,19)(4,24)(5,23)(6,22)(7,15)(8,14)(9,13)(10,18)(11,17)(12,16) );
G=PermutationGroup([[(1,3,5),(2,4,6),(7,11,9),(8,12,10),(13,15,17),(14,16,18),(19,23,21),(20,24,22)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,10,4,7),(2,9,5,12),(3,8,6,11),(13,23,16,20),(14,22,17,19),(15,21,18,24)], [(1,5,3),(2,6,4),(7,12,11,10,9,8),(13,18,17,16,15,14),(19,23,21),(20,24,22)], [(1,21),(2,20),(3,19),(4,24),(5,23),(6,22),(7,15),(8,14),(9,13),(10,18),(11,17),(12,16)]])
G:=TransitiveGroup(24,1282);
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | ··· | 3H | 3I | 3J | 3K | 4A | 4B | 6A | 6B | 6C | ··· | 6J | 6K | ··· | 6Y | 6Z | 6AA | 6AB | 6AC | 6AD | 6AE | 6AF | 6AG | 6AH | ··· | 6AM | 6AN | 6AO | 12A | 12B | 12C | 12D | 12E | ··· | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 6 | 6 | 9 | 9 | 18 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | 18 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | S3 | D4 | D6 | D6 | D6 | C3×S3 | C3×D4 | S3×C6 | S3×C6 | S3×C6 | S32 | S3×D4 | C2×S32 | C3×S32 | Dic3⋊D6 | C3×S3×D4 | S32×C6 | C3×Dic3⋊D6 |
| kernel | C3×Dic3⋊D6 | C3×C6.D6 | C3×C3⋊D12 | C32×C3⋊D4 | S32×C6 | C2×C6×C3⋊S3 | Dic3⋊D6 | C6.D6 | C3⋊D12 | C3×C3⋊D4 | C2×S32 | C22×C3⋊S3 | C3×C3⋊D4 | C3×C3⋊S3 | C3×Dic3 | S3×C6 | C62 | C3⋊D4 | C3⋊S3 | Dic3 | D6 | C2×C6 | C2×C6 | C32 | C6 | C22 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 4 |
Matrix representation of C3×Dic3⋊D6 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 3 | 6 | 5 | 6 |
| 4 | 4 | 1 | 3 |
| 1 | 1 | 3 | 5 |
| 1 | 6 | 3 | 6 |
| 4 | 0 | 5 | 0 |
| 4 | 6 | 4 | 6 |
| 5 | 0 | 3 | 0 |
| 4 | 2 | 0 | 1 |
| 5 | 4 | 6 | 1 |
| 0 | 4 | 5 | 6 |
| 3 | 0 | 3 | 4 |
| 6 | 6 | 6 | 2 |
| 6 | 3 | 1 | 5 |
| 2 | 2 | 2 | 0 |
| 6 | 6 | 4 | 2 |
| 6 | 1 | 1 | 2 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[3,4,1,1,6,4,1,6,5,1,3,3,6,3,5,6],[4,4,5,4,0,6,0,2,5,4,3,0,0,6,0,1],[5,0,3,6,4,4,0,6,6,5,3,6,1,6,4,2],[6,2,6,6,3,2,6,1,1,2,4,1,5,0,2,2] >;
C3×Dic3⋊D6 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_3\rtimes D_6 % in TeX
G:=Group("C3xDic3:D6"); // GroupNames label
G:=SmallGroup(432,659);
// by ID
G=gap.SmallGroup(432,659);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,590,303,2028,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^6=d^6=e^2=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,d*c*d^-1=b^3*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations