Aliases: He3.3C12, C9○He3⋊C4, He3⋊C4.C3, C9.(C32⋊C4), He3⋊C2.3C6, He3.4C6.C2, C3.3(C3×C32⋊C4), SmallGroup(324,111)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — He3.3C12 |
Generators and relations for He3.3C12
G = < a,b,c,d | a3=b3=c3=1, d12=b, ab=ba, cac-1=ab-1, dad-1=abc, bc=cb, bd=db, dcd-1=ac-1 >
Smallest permutation representation of He3.3C12
►On 54 points
Generators in S54
(1 51 21)(2 40 46)(3 35 41)(4 24 30)(5 19 25)(6 44 50)(7 39 45)(8 28 34)(9 23 29)(10 48 54)(11 43 49)(12 32 38)(13 27 33)(14 52 22)(15 47 53)(16 36 42)(17 31 37)(18 20 26)
(1 13 7)(2 14 8)(3 15 9)(4 16 10)(5 17 11)(6 18 12)(19 31 43)(20 32 44)(21 33 45)(22 34 46)(23 35 47)(24 36 48)(25 37 49)(26 38 50)(27 39 51)(28 40 52)(29 41 53)(30 42 54)
(2 40 22)(4 24 42)(6 44 26)(8 28 46)(10 48 30)(12 32 50)(14 52 34)(16 36 54)(18 20 38)(19 31 43)(21 45 33)(23 35 47)(25 49 37)(27 39 51)(29 53 41)
(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 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54)
G:=sub<Sym(54)| (1,51,21)(2,40,46)(3,35,41)(4,24,30)(5,19,25)(6,44,50)(7,39,45)(8,28,34)(9,23,29)(10,48,54)(11,43,49)(12,32,38)(13,27,33)(14,52,22)(15,47,53)(16,36,42)(17,31,37)(18,20,26), (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)(19,31,43)(20,32,44)(21,33,45)(22,34,46)(23,35,47)(24,36,48)(25,37,49)(26,38,50)(27,39,51)(28,40,52)(29,41,53)(30,42,54), (2,40,22)(4,24,42)(6,44,26)(8,28,46)(10,48,30)(12,32,50)(14,52,34)(16,36,54)(18,20,38)(19,31,43)(21,45,33)(23,35,47)(25,49,37)(27,39,51)(29,53,41), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)>;
G:=Group( (1,51,21)(2,40,46)(3,35,41)(4,24,30)(5,19,25)(6,44,50)(7,39,45)(8,28,34)(9,23,29)(10,48,54)(11,43,49)(12,32,38)(13,27,33)(14,52,22)(15,47,53)(16,36,42)(17,31,37)(18,20,26), (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)(19,31,43)(20,32,44)(21,33,45)(22,34,46)(23,35,47)(24,36,48)(25,37,49)(26,38,50)(27,39,51)(28,40,52)(29,41,53)(30,42,54), (2,40,22)(4,24,42)(6,44,26)(8,28,46)(10,48,30)(12,32,50)(14,52,34)(16,36,54)(18,20,38)(19,31,43)(21,45,33)(23,35,47)(25,49,37)(27,39,51)(29,53,41), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54) );
G=PermutationGroup([[(1,51,21),(2,40,46),(3,35,41),(4,24,30),(5,19,25),(6,44,50),(7,39,45),(8,28,34),(9,23,29),(10,48,54),(11,43,49),(12,32,38),(13,27,33),(14,52,22),(15,47,53),(16,36,42),(17,31,37),(18,20,26)], [(1,13,7),(2,14,8),(3,15,9),(4,16,10),(5,17,11),(6,18,12),(19,31,43),(20,32,44),(21,33,45),(22,34,46),(23,35,47),(24,36,48),(25,37,49),(26,38,50),(27,39,51),(28,40,52),(29,41,53),(30,42,54)], [(2,40,22),(4,24,42),(6,44,26),(8,28,46),(10,48,30),(12,32,50),(14,52,34),(16,36,54),(18,20,38),(19,31,43),(21,45,33),(23,35,47),(25,49,37),(27,39,51),(29,53,41)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)]])
42 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 9A | ··· | 9F | 9G | 9H | 9I | 9J | 12A | 12B | 12C | 12D | 18A | ··· | 18F | 36A | ··· | 36L |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 9 | ··· | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 9 | 1 | 1 | 12 | 12 | 9 | 9 | 9 | 9 | 1 | ··· | 1 | 12 | 12 | 12 | 12 | 9 | 9 | 9 | 9 | 9 | ··· | 9 | 9 | ··· | 9 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 4 |
| type | + | + | + | ||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | He3.3C12 | C32⋊C4 | C3×C32⋊C4 |
| kernel | He3.3C12 | He3.4C6 | He3⋊C4 | C9○He3 | He3⋊C2 | He3 | C1 | C9 | C3 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 24 | 2 | 4 |
Matrix representation of He3.3C12 ►in GL3(𝔽37) generated by
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 10 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 10 |
| 1 | 0 | 0 |
| 0 | 26 | 0 |
| 0 | 0 | 10 |
| 17 | 17 | 22 |
| 17 | 22 | 17 |
| 35 | 22 | 22 |
G:=sub<GL(3,GF(37))| [0,0,1,1,0,0,0,1,0],[10,0,0,0,10,0,0,0,10],[1,0,0,0,26,0,0,0,10],[17,17,35,17,22,22,22,17,22] >;
He3.3C12 in GAP, Magma, Sage, TeX
{\rm He}_3._3C_{12} % in TeX
G:=Group("He3.3C12"); // GroupNames label
G:=SmallGroup(324,111);
// by ID
G=gap.SmallGroup(324,111);
# by ID
G:=PCGroup([6,-2,-3,-2,-3,3,-3,36,655,2019,111,2884,916,382]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=1,d^12=b,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a*b*c,b*c=c*b,b*d=d*b,d*c*d^-1=a*c^-1>;
// generators/relations
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