direct product, metabelian, supersoluble, monomial
Aliases: Dic3×He3, C33⋊3C12, C3⋊(C4×He3), C6.(C2×He3), C2.(S3×He3), (C3×He3)⋊4C4, (C6×He3).2C2, C6.9(S3×C32), (C32×C6).5C6, (C32×Dic3)⋊C3, (C2×He3).14S3, C32.7(C3×C12), C32⋊5(C3×Dic3), C3.5(C32×Dic3), (C3×Dic3).2C32, (C3×C6).16(C3×C6), (C3×C6).22(C3×S3), SmallGroup(324,93)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3×He3
G = < a,b,c,d,e | a6=c3=d3=e3=1, b2=a3, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >
Subgroups: 284 in 100 conjugacy classes, 35 normal (15 characteristic)
C1, C2, C3, C3, C4, C6, C6, C32, C32, C32, Dic3, C12, C3×C6, C3×C6, C3×C6, He3, He3, C33, C3×Dic3, C3×Dic3, C3×C12, C2×He3, C2×He3, C32×C6, C3×He3, C4×He3, C32×Dic3, C6×He3, Dic3×He3
Quotients: C1, C2, C3, C4, S3, C6, C32, Dic3, C12, C3×S3, C3×C6, He3, C3×Dic3, C3×C12, C2×He3, S3×C32, C4×He3, C32×Dic3, S3×He3, Dic3×He3
►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 22 4 19)(2 21 5 24)(3 20 6 23)(7 26 10 29)(8 25 11 28)(9 30 12 27)(13 32 16 35)(14 31 17 34)(15 36 18 33)
(1 17 11)(2 18 12)(3 13 7)(4 14 8)(5 15 9)(6 16 10)(19 31 25)(20 32 26)(21 33 27)(22 34 28)(23 35 29)(24 36 30)
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 35 33)(32 36 34)
(1 11 15)(2 12 16)(3 7 17)(4 8 18)(5 9 13)(6 10 14)(19 25 33)(20 26 34)(21 27 35)(22 28 36)(23 29 31)(24 30 32)
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), (1,22,4,19)(2,21,5,24)(3,20,6,23)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,32,16,35)(14,31,17,34)(15,36,18,33), (1,17,11)(2,18,12)(3,13,7)(4,14,8)(5,15,9)(6,16,10)(19,31,25)(20,32,26)(21,33,27)(22,34,28)(23,35,29)(24,36,30), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(31,35,33)(32,36,34), (1,11,15)(2,12,16)(3,7,17)(4,8,18)(5,9,13)(6,10,14)(19,25,33)(20,26,34)(21,27,35)(22,28,36)(23,29,31)(24,30,32)>;
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), (1,22,4,19)(2,21,5,24)(3,20,6,23)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,32,16,35)(14,31,17,34)(15,36,18,33), (1,17,11)(2,18,12)(3,13,7)(4,14,8)(5,15,9)(6,16,10)(19,31,25)(20,32,26)(21,33,27)(22,34,28)(23,35,29)(24,36,30), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(31,35,33)(32,36,34), (1,11,15)(2,12,16)(3,7,17)(4,8,18)(5,9,13)(6,10,14)(19,25,33)(20,26,34)(21,27,35)(22,28,36)(23,29,31)(24,30,32) );
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)], [(1,22,4,19),(2,21,5,24),(3,20,6,23),(7,26,10,29),(8,25,11,28),(9,30,12,27),(13,32,16,35),(14,31,17,34),(15,36,18,33)], [(1,17,11),(2,18,12),(3,13,7),(4,14,8),(5,15,9),(6,16,10),(19,31,25),(20,32,26),(21,33,27),(22,34,28),(23,35,29),(24,36,30)], [(1,3,5),(2,4,6),(7,9,11),(8,10,12),(13,15,17),(14,16,18),(19,23,21),(20,24,22),(25,29,27),(26,30,28),(31,35,33),(32,36,34)], [(1,11,15),(2,12,16),(3,7,17),(4,8,18),(5,9,13),(6,10,14),(19,25,33),(20,26,34),(21,27,35),(22,28,36),(23,29,31),(24,30,32)]])
66 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | ··· | 3M | 3N | ··· | 3U | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | ··· | 6M | 6N | ··· | 6U | 12A | 12B | 12C | 12D | 12E | ··· | 12T |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 3 | 3 | 3 | 3 | 9 | ··· | 9 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 6 | 6 |
| type | + | + | + | - | |||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | S3 | Dic3 | C3×S3 | C3×Dic3 | He3 | C2×He3 | C4×He3 | S3×He3 | Dic3×He3 |
| kernel | Dic3×He3 | C6×He3 | C32×Dic3 | C3×He3 | C32×C6 | C33 | C2×He3 | He3 | C3×C6 | C32 | Dic3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 8 | 2 | 8 | 16 | 1 | 1 | 8 | 8 | 2 | 2 | 4 | 2 | 2 |
Matrix representation of Dic3×He3 ►in GL5(𝔽13)
| 1 | 1 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 5 | 0 | 0 | 0 | 0 |
| 8 | 8 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
G:=sub<GL(5,GF(13))| [1,12,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[5,8,0,0,0,0,8,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[9,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,9,0,0] >;
Dic3×He3 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times {\rm He}_3 % in TeX
G:=Group("Dic3xHe3"); // GroupNames label
G:=SmallGroup(324,93);
// by ID
G=gap.SmallGroup(324,93);
# by ID
G:=PCGroup([6,-2,-3,-3,-2,-3,-3,108,386,7781]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=c^3=d^3=e^3=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c*d^-1,d*e=e*d>;
// generators/relations