non-abelian, supersoluble, monomial
Aliases: C62⋊2D6, He3⋊7(C2×D4), C3⋊Dic3⋊2D6, C32⋊4(S3×D4), He3⋊3D4⋊6C2, He3⋊6D4⋊4C2, He3⋊C2⋊3D4, C32⋊7D4⋊4S3, C3.2(Dic3⋊D6), C32⋊C12⋊2C22, C22⋊3(C32⋊D6), (C2×He3).18C23, (C22×He3)⋊2C22, C6.92(C2×S32), (C2×C3⋊S3)⋊3D6, (C2×C6).58S32, He3⋊(C2×C4)⋊3C2, (C2×C32⋊D6)⋊4C2, C2.18(C2×C32⋊D6), (C2×C32⋊C6)⋊3C22, (C3×C6).18(C22×S3), (C22×He3⋊C2)⋊2C2, (C2×He3⋊C2)⋊4C22, SmallGroup(432,324)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62⋊2D6
G = < a,b,c,d | a6=b6=c6=d2=1, ab=ba, cac-1=a-1b, dad=a-1b4, cbc-1=b-1, bd=db, dcd=c-1 >
Subgroups: 1467 in 221 conjugacy classes, 37 normal (15 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C22×S3, C22×C6, He3, C3×Dic3, C3⋊Dic3, S32, S3×C6, C2×C3⋊S3, C62, C62, S3×D4, C2×C3⋊D4, C32⋊C6, He3⋊C2, He3⋊C2, C2×He3, C2×He3, S3×Dic3, D6⋊S3, C3⋊D12, C3×C3⋊D4, C32⋊7D4, C2×S32, S3×C2×C6, C32⋊C12, C32⋊D6, C2×C32⋊C6, C2×He3⋊C2, C2×He3⋊C2, C22×He3, S3×C3⋊D4, He3⋊(C2×C4), He3⋊3D4, He3⋊6D4, C2×C32⋊D6, C22×He3⋊C2, C62⋊2D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, S32, S3×D4, C2×S32, C32⋊D6, Dic3⋊D6, C2×C32⋊D6, C62⋊2D6
►On 36 points
(7 8)(9 10)(11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27 28 29 30)(31 32 33 34 35 36)
(1 2 6 5 4 3)(7 9 12 8 10 11)(13 17 14 18 15 16)(19 23 21 22 20 24)(25 30 29 28 27 26)(31 32 33 34 35 36)
(1 28 22 8 18 32)(2 29 20 12 15 31)(3 27 21 10 14 33)(4 26 23 11 17 34)(5 25 19 7 13 35)(6 30 24 9 16 36)
(1 22)(2 20)(3 21)(4 23)(5 19)(6 24)(7 35)(8 32)(9 36)(10 33)(11 34)(12 31)
G:=sub<Sym(36)| (7,8)(9,10)(11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36), (1,2,6,5,4,3)(7,9,12,8,10,11)(13,17,14,18,15,16)(19,23,21,22,20,24)(25,30,29,28,27,26)(31,32,33,34,35,36), (1,28,22,8,18,32)(2,29,20,12,15,31)(3,27,21,10,14,33)(4,26,23,11,17,34)(5,25,19,7,13,35)(6,30,24,9,16,36), (1,22)(2,20)(3,21)(4,23)(5,19)(6,24)(7,35)(8,32)(9,36)(10,33)(11,34)(12,31)>;
G:=Group( (7,8)(9,10)(11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36), (1,2,6,5,4,3)(7,9,12,8,10,11)(13,17,14,18,15,16)(19,23,21,22,20,24)(25,30,29,28,27,26)(31,32,33,34,35,36), (1,28,22,8,18,32)(2,29,20,12,15,31)(3,27,21,10,14,33)(4,26,23,11,17,34)(5,25,19,7,13,35)(6,30,24,9,16,36), (1,22)(2,20)(3,21)(4,23)(5,19)(6,24)(7,35)(8,32)(9,36)(10,33)(11,34)(12,31) );
G=PermutationGroup([[(7,8),(9,10),(11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24),(25,26,27,28,29,30),(31,32,33,34,35,36)], [(1,2,6,5,4,3),(7,9,12,8,10,11),(13,17,14,18,15,16),(19,23,21,22,20,24),(25,30,29,28,27,26),(31,32,33,34,35,36)], [(1,28,22,8,18,32),(2,29,20,12,15,31),(3,27,21,10,14,33),(4,26,23,11,17,34),(5,25,19,7,13,35),(6,30,24,9,16,36)], [(1,22),(2,20),(3,21),(4,23),(5,19),(6,24),(7,35),(8,32),(9,36),(10,33),(11,34),(12,31)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | ··· | 6J | 6K | 6L | 6M | 6N | 6O | 6P | 12A | 12B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 |
| size | 1 | 1 | 2 | 9 | 9 | 18 | 18 | 18 | 2 | 6 | 6 | 12 | 18 | 18 | 2 | 2 | 2 | 6 | 6 | 12 | ··· | 12 | 18 | 18 | 18 | 18 | 36 | 36 | 36 | 36 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | S32 | S3×D4 | C2×S32 | Dic3⋊D6 | C32⋊D6 | C2×C32⋊D6 | C62⋊2D6 |
| kernel | C62⋊2D6 | He3⋊(C2×C4) | He3⋊3D4 | He3⋊6D4 | C2×C32⋊D6 | C22×He3⋊C2 | C32⋊7D4 | He3⋊C2 | C3⋊Dic3 | C2×C3⋊S3 | C62 | C2×C6 | C32 | C6 | C3 | C22 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of C62⋊2D6 ►in GL6(𝔽13)
| 1 | 0 | 12 | 12 | 0 | 7 |
| 0 | 3 | 9 | 0 | 7 | 6 |
| 0 | 0 | 9 | 0 | 0 | 2 |
| 0 | 0 | 0 | 12 | 0 | 10 |
| 0 | 0 | 0 | 0 | 4 | 9 |
| 0 | 0 | 0 | 0 | 0 | 10 |
| 4 | 0 | 0 | 3 | 0 | 0 |
| 0 | 4 | 0 | 0 | 3 | 0 |
| 0 | 0 | 4 | 0 | 0 | 12 |
| 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 7 | 9 | 0 | 12 | 1 |
| 0 | 2 | 0 | 0 | 12 | 0 |
| 11 | 2 | 0 | 1 | 12 | 0 |
| 0 | 10 | 11 | 0 | 8 | 4 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 8 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 1 | 0 |
| 0 | 0 | 0 | 8 | 0 | 1 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,3,0,0,0,0,12,9,9,0,0,0,12,0,0,12,0,0,0,7,0,0,4,0,7,6,2,10,9,10],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,3,0,0,10,0,0,0,3,0,0,10,0,0,0,12,0,0,10],[0,12,0,0,11,0,1,1,7,2,2,10,0,0,9,0,0,11,0,0,0,0,1,0,0,0,12,12,12,8,0,0,1,0,0,4],[12,12,8,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,12,12,12,8,0,0,0,0,1,0,0,0,0,0,0,1] >;
C62⋊2D6 in GAP, Magma, Sage, TeX
C_6^2\rtimes_2D_6
% in TeX
G:=Group("C6^2:2D6"); // GroupNames label
G:=SmallGroup(432,324);
// by ID
G=gap.SmallGroup(432,324);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,135,571,4037,537,14118,7069]);
// 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,d*a*d=a^-1*b^4,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^-1>;
// generators/relations