direct product, metabelian, supersoluble, monomial, rational
Aliases: C2×Dic3⋊D6, C62⋊3C23, C23⋊5S32, C6⋊3(S3×D4), C3⋊D4⋊9D6, (C22×C6)⋊8D6, (S3×C6)⋊5C23, D6⋊4(C22×S3), C32⋊8(C22×D4), (C2×Dic3)⋊16D6, (C22×S3)⋊13D6, C6.36(S3×C23), (C3×C6).36C24, (C2×C62)⋊7C22, (C6×Dic3)⋊9C22, (C3×Dic3)⋊3C23, Dic3⋊3(C22×S3), C3⋊D12⋊18C22, C6.D6⋊12C22, C3⋊4(C2×S3×D4), C3⋊S3⋊4(C2×D4), C22⋊4(C2×S32), (C3×C6)⋊7(C2×D4), (C2×C3⋊S3)⋊18D4, (C22×S32)⋊9C2, (C6×C3⋊D4)⋊7C2, (C2×S32)⋊12C22, (C2×C3⋊D4)⋊11S3, (C23×C3⋊S3)⋊5C2, (C2×C3⋊S3)⋊5C23, (S3×C2×C6)⋊12C22, (C2×C6)⋊5(C22×S3), C2.36(C22×S32), (C2×C6.D6)⋊6C2, (C2×C3⋊D12)⋊22C2, (C3×C3⋊D4)⋊13C22, (C22×C3⋊S3)⋊14C22, SmallGroup(288,977)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Dic3⋊D6
G = < a,b,c,d,e | a2=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: 2370 in 539 conjugacy classes, 124 normal (14 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C22×C4, C2×D4, C24, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, C22×D4, C3×Dic3, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, S3×C2×C4, C2×D12, S3×D4, C2×C3⋊D4, C2×C3⋊D4, C6×D4, S3×C23, C6.D6, C3⋊D12, C6×Dic3, C3×C3⋊D4, C2×S32, C2×S32, S3×C2×C6, C22×C3⋊S3, C22×C3⋊S3, C22×C3⋊S3, C2×C62, C2×S3×D4, C2×C6.D6, C2×C3⋊D12, Dic3⋊D6, C6×C3⋊D4, C22×S32, C23×C3⋊S3, C2×Dic3⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, S32, S3×D4, S3×C23, C2×S32, C2×S3×D4, Dic3⋊D6, C22×S32, C2×Dic3⋊D6
►On 24 points - transitive group 24T671
(1 11)(2 12)(3 7)(4 8)(5 9)(6 10)(13 19)(14 20)(15 21)(16 22)(17 23)(18 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 22 4 19)(2 21 5 24)(3 20 6 23)(7 14 10 17)(8 13 11 16)(9 18 12 15)
(1 7 5 11 3 9)(2 8 6 12 4 10)(13 20 15 22 17 24)(14 21 16 23 18 19)
(1 5)(2 4)(8 12)(9 11)(13 15)(16 18)(19 21)(22 24)
G:=sub<Sym(24)| (1,11)(2,12)(3,7)(4,8)(5,9)(6,10)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,22,4,19)(2,21,5,24)(3,20,6,23)(7,14,10,17)(8,13,11,16)(9,18,12,15), (1,7,5,11,3,9)(2,8,6,12,4,10)(13,20,15,22,17,24)(14,21,16,23,18,19), (1,5)(2,4)(8,12)(9,11)(13,15)(16,18)(19,21)(22,24)>;
G:=Group( (1,11)(2,12)(3,7)(4,8)(5,9)(6,10)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,22,4,19)(2,21,5,24)(3,20,6,23)(7,14,10,17)(8,13,11,16)(9,18,12,15), (1,7,5,11,3,9)(2,8,6,12,4,10)(13,20,15,22,17,24)(14,21,16,23,18,19), (1,5)(2,4)(8,12)(9,11)(13,15)(16,18)(19,21)(22,24) );
G=PermutationGroup([[(1,11),(2,12),(3,7),(4,8),(5,9),(6,10),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,22,4,19),(2,21,5,24),(3,20,6,23),(7,14,10,17),(8,13,11,16),(9,18,12,15)], [(1,7,5,11,3,9),(2,8,6,12,4,10),(13,20,15,22,17,24),(14,21,16,23,18,19)], [(1,5),(2,4),(8,12),(9,11),(13,15),(16,18),(19,21),(22,24)]])
G:=TransitiveGroup(24,671);
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | ··· | 6Q | 6R | 6S | 6T | 6U | 12A | 12B | 12C | 12D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 18 | 18 | 2 | 2 | 4 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | S32 | S3×D4 | C2×S32 | Dic3⋊D6 |
| kernel | C2×Dic3⋊D6 | C2×C6.D6 | C2×C3⋊D12 | Dic3⋊D6 | C6×C3⋊D4 | C22×S32 | C23×C3⋊S3 | C2×C3⋊D4 | C2×C3⋊S3 | C2×Dic3 | C3⋊D4 | C22×S3 | C22×C6 | C23 | C6 | C22 | C2 |
| # reps | 1 | 1 | 2 | 8 | 2 | 1 | 1 | 2 | 4 | 2 | 8 | 2 | 2 | 1 | 4 | 3 | 4 |
Matrix representation of C2×Dic3⋊D6 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 5 | 0 | 0 |
| 0 | 0 | 10 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 3 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,10,0,0,0,0,5,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,3,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C2×Dic3⋊D6 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_3\rtimes D_6 % in TeX
G:=Group("C2xDic3:D6"); // GroupNames label
G:=SmallGroup(288,977);
// by ID
G=gap.SmallGroup(288,977);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,675,346,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=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