non-abelian, soluble, monomial
Aliases: C62⋊6Dic3, (C3×C6).9S4, (C6×A4).1S3, C32⋊A4⋊2C4, C6.12(C3⋊S4), (C3×A4)⋊1Dic3, (C2×C62).9S3, C32⋊2(A4⋊C4), C22⋊(He3⋊3C4), C3.3(C6.7S4), C23.(He3⋊C2), C2.1(C32⋊S4), (C2×C32⋊A4).2C2, (C22×C6).1(C3⋊S3), (C2×C6).1(C3⋊Dic3), SmallGroup(432,260)
Series: Derived ►Chief ►Lower central ►Upper central
| C32⋊A4 — C62⋊6Dic3 |
Generators and relations for C62⋊6Dic3
G = < a,b,c,d | a6=b6=c6=1, d2=c3, ab=ba, cac-1=ab-1, dad-1=a2b3, cbc-1=dbd-1=a3b4, dcd-1=c-1 >
Subgroups: 489 in 98 conjugacy classes, 21 normal (15 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C6, C6, C2×C4, C23, C32, C32, Dic3, C12, A4, C2×C6, C2×C6, C22⋊C4, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×A4, C22×C6, C22×C6, He3, C3×Dic3, C3×A4, C62, C62, C6.D4, C3×C22⋊C4, A4⋊C4, C2×He3, C6×Dic3, C6×A4, C2×C62, He3⋊3C4, C32⋊A4, C3×C6.D4, C3×A4⋊C4, C2×C32⋊A4, C62⋊6Dic3
Quotients: C1, C2, C4, S3, Dic3, C3⋊S3, S4, C3⋊Dic3, A4⋊C4, He3⋊C2, C3⋊S4, He3⋊3C4, C6.7S4, C32⋊S4, C62⋊6Dic3
►On 36 points
(1 2)(3 4)(5 6)(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 4 7 2 3 8)(5 11 10 6 12 9)(13 23 14 24 15 22)(16 21 17 19 18 20)(25 29 27)(26 30 28)(31 35 33)(32 36 34)
(1 17 30 2 20 27)(3 16 28 4 19 25)(5 13 35 6 24 32)(7 18 26 8 21 29)(9 22 34 10 14 31)(11 23 36 12 15 33)
(1 6 2 5)(3 11 4 12)(7 9 8 10)(13 27 24 30)(14 29 22 26)(15 25 23 28)(16 33 19 36)(17 35 20 32)(18 31 21 34)
G:=sub<Sym(36)| (1,2)(3,4)(5,6)(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), (1,4,7,2,3,8)(5,11,10,6,12,9)(13,23,14,24,15,22)(16,21,17,19,18,20)(25,29,27)(26,30,28)(31,35,33)(32,36,34), (1,17,30,2,20,27)(3,16,28,4,19,25)(5,13,35,6,24,32)(7,18,26,8,21,29)(9,22,34,10,14,31)(11,23,36,12,15,33), (1,6,2,5)(3,11,4,12)(7,9,8,10)(13,27,24,30)(14,29,22,26)(15,25,23,28)(16,33,19,36)(17,35,20,32)(18,31,21,34)>;
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)(25,26,27,28,29,30)(31,32,33,34,35,36), (1,4,7,2,3,8)(5,11,10,6,12,9)(13,23,14,24,15,22)(16,21,17,19,18,20)(25,29,27)(26,30,28)(31,35,33)(32,36,34), (1,17,30,2,20,27)(3,16,28,4,19,25)(5,13,35,6,24,32)(7,18,26,8,21,29)(9,22,34,10,14,31)(11,23,36,12,15,33), (1,6,2,5)(3,11,4,12)(7,9,8,10)(13,27,24,30)(14,29,22,26)(15,25,23,28)(16,33,19,36)(17,35,20,32)(18,31,21,34) );
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),(25,26,27,28,29,30),(31,32,33,34,35,36)], [(1,4,7,2,3,8),(5,11,10,6,12,9),(13,23,14,24,15,22),(16,21,17,19,18,20),(25,29,27),(26,30,28),(31,35,33),(32,36,34)], [(1,17,30,2,20,27),(3,16,28,4,19,25),(5,13,35,6,24,32),(7,18,26,8,21,29),(9,22,34,10,14,31),(11,23,36,12,15,33)], [(1,6,2,5),(3,11,4,12),(7,9,8,10),(13,27,24,30),(14,29,22,26),(15,25,23,28),(16,33,19,36),(17,35,20,32),(18,31,21,34)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | ··· | 6M | 6N | 6O | 6P | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 6 | 24 | 24 | 24 | 18 | 18 | 18 | 18 | 1 | 1 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 24 | 24 | 24 | 18 | ··· | 18 |
38 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | - | - | + | + | - | ||||||||
| image | C1 | C2 | C4 | S3 | S3 | Dic3 | Dic3 | S4 | A4⋊C4 | He3⋊C2 | He3⋊3C4 | C32⋊S4 | C62⋊6Dic3 | C3⋊S4 | C6.7S4 | C32⋊S4 | C62⋊6Dic3 |
| kernel | C62⋊6Dic3 | C2×C32⋊A4 | C32⋊A4 | C6×A4 | C2×C62 | C3×A4 | C62 | C3×C6 | C32 | C23 | C22 | C2 | C1 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 3 | 1 | 3 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of C62⋊6Dic3 ►in GL3(𝔽13) generated by
| 12 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 9 |
| 10 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 10 |
| 0 | 12 | 0 |
| 0 | 0 | 12 |
| 12 | 0 | 0 |
| 8 | 0 | 0 |
| 0 | 0 | 8 |
| 0 | 8 | 0 |
G:=sub<GL(3,GF(13))| [12,0,0,0,10,0,0,0,9],[10,0,0,0,3,0,0,0,10],[0,0,12,12,0,0,0,12,0],[8,0,0,0,0,8,0,8,0] >;
C62⋊6Dic3 in GAP, Magma, Sage, TeX
C_6^2\rtimes_6{\rm Dic}_3 % in TeX
G:=Group("C6^2:6Dic3"); // GroupNames label
G:=SmallGroup(432,260);
// by ID
G=gap.SmallGroup(432,260);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,14,170,675,353,9077,2287,5298,3989]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^6=1,d^2=c^3,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^2*b^3,c*b*c^-1=d*b*d^-1=a^3*b^4,d*c*d^-1=c^-1>;
// generators/relations