non-abelian, supersoluble, monomial
Aliases: He3:C12, C33:1Dic3, C2.C3wrS3, C3wrC3:2C4, He3:3C4:1C3, (C2xHe3).1C6, (C32xC6).2S3, C3.6(C32:C12), C6.11(C32:C6), C32.1(C3xDic3), (C3xC6).1(C3xS3), (C2xC3wrC3).2C2, SmallGroup(324,13)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — He3:C12 |
Generators and relations for He3:C12
G = < a,b,c,d | a3=b3=c3=d12=1, ab=ba, cac-1=ab-1, dad-1=a-1b, bc=cb, bd=db, dcd-1=ac-1 >
Quotients: C1, C2, C3, C4, S3, C6, Dic3, C12, C3xS3, C3xDic3, C32:C6, C32:C12, C3wrS3, He3:C12
3C3
3C3
3C3
3C3
9C3
9C4
3C6
3C6
3C6
3C6
9C6
3C32
3C32
3C32
3C32
3C32
6C9
3Dic3
9C12
9C12
9C12
9C12
9Dic3
6C18
Smallest permutation representation of He3:C12
►On 36 points
Generators in S36
(2 10 6)(4 12 8)(14 22 18)(16 24 20)(25 33 29)(26 34 30)(27 35 31)(28 36 32)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)
(1 28 20)(2 21 33)(3 26 22)(4 23 31)(5 36 24)(6 13 29)(7 34 14)(8 15 27)(9 32 16)(10 17 25)(11 30 18)(12 19 35)
(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)
G:=sub<Sym(36)| (2,10,6)(4,12,8)(14,22,18)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36), (1,28,20)(2,21,33)(3,26,22)(4,23,31)(5,36,24)(6,13,29)(7,34,14)(8,15,27)(9,32,16)(10,17,25)(11,30,18)(12,19,35), (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)>;
G:=Group( (2,10,6)(4,12,8)(14,22,18)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36), (1,28,20)(2,21,33)(3,26,22)(4,23,31)(5,36,24)(6,13,29)(7,34,14)(8,15,27)(9,32,16)(10,17,25)(11,30,18)(12,19,35), (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) );
G=PermutationGroup([[(2,10,6),(4,12,8),(14,22,18),(16,24,20),(25,33,29),(26,34,30),(27,35,31),(28,36,32)], [(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,29,33),(26,30,34),(27,31,35),(28,32,36)], [(1,28,20),(2,21,33),(3,26,22),(4,23,31),(5,36,24),(6,13,29),(7,34,14),(8,15,27),(9,32,16),(10,17,25),(11,30,18),(12,19,35)], [(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)]])
44 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3H | 3I | 3J | 4A | 4B | 6A | 6B | 6C | ··· | 6H | 6I | 6J | 9A | 9B | 12A | ··· | 12P | 18A | 18B |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 9 | 9 | 12 | ··· | 12 | 18 | 18 |
| size | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 6 | 18 | 9 | 9 | 1 | 1 | 3 | ··· | 3 | 6 | 18 | 18 | 18 | 9 | ··· | 9 | 18 | 18 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | 6 |
| type | + | + | + | - | + | - | ||||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | S3 | Dic3 | C3xS3 | C3xDic3 | C3wrS3 | He3:C12 | C32:C6 | C32:C12 |
| kernel | He3:C12 | C2xC3wrC3 | He3:3C4 | C3wrC3 | C2xHe3 | He3 | C32xC6 | C33 | C3xC6 | C32 | C2 | C1 | C6 | C3 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 12 | 12 | 1 | 1 |
Matrix representation of He3:C12 ►in GL3(F13) generated by
| 3 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 9 |
| 9 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 9 |
| 0 | 4 | 0 |
| 0 | 0 | 7 |
| 7 | 0 | 0 |
| 11 | 0 | 0 |
| 0 | 0 | 4 |
| 0 | 3 | 0 |
G:=sub<GL(3,GF(13))| [3,0,0,0,1,0,0,0,9],[9,0,0,0,9,0,0,0,9],[0,0,7,4,0,0,0,7,0],[11,0,0,0,0,3,0,4,0] >;
He3:C12 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes C_{12} % in TeX
G:=Group("He3:C12"); // GroupNames label
G:=SmallGroup(324,13);
// by ID
G=gap.SmallGroup(324,13);
# by ID
G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,579,585,5404,382]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=d^12=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1*b,b*c=c*b,b*d=d*b,d*c*d^-1=a*c^-1>;
// generators/relations
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