non-abelian, supersoluble, monomial
Aliases: He3⋊6Dic3, C33⋊6Dic3, (C3×He3)⋊7C4, C3⋊(He3⋊3C4), (C6×He3).5C2, C2.(He3⋊5S3), (C2×He3).12S3, (C32×C6).12S3, C3.2(C33⋊5C4), C6.4(C33⋊C2), C32⋊2(C3⋊Dic3), C6.11(He3⋊C2), (C3×C6).12(C3⋊S3), SmallGroup(324,104)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×He3 — He3⋊6Dic3 |
Generators and relations for He3⋊6Dic3
G = < a,b,c,d,e | a3=b3=c3=d6=1, e2=d3, ab=ba, cac-1=dad-1=ab-1, eae-1=a-1, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >
Subgroups: 548 in 148 conjugacy classes, 61 normal (11 characteristic)
C1, C2, C3, C3, C4, C6, C6, C32, C32, C32, Dic3, C12, C3×C6, C3×C6, C3×C6, He3, C33, C3×Dic3, C3⋊Dic3, C2×He3, C32×C6, C3×He3, He3⋊3C4, C3×C3⋊Dic3, C6×He3, He3⋊6Dic3
Quotients: C1, C2, C4, S3, Dic3, C3⋊S3, C3⋊Dic3, He3⋊C2, C33⋊C2, He3⋊3C4, C33⋊5C4, He3⋊5S3, He3⋊6Dic3
►On 36 points
(1 3 27)(2 23 19)(4 6 30)(5 20 22)(7 35 31)(8 13 12)(9 11 16)(10 32 34)(14 33 18)(15 17 36)(21 26 28)(24 29 25)
(1 29 20)(2 30 21)(3 25 22)(4 26 23)(5 27 24)(6 28 19)(7 16 33)(8 17 34)(9 18 35)(10 13 36)(11 14 31)(12 15 32)
(1 27 22)(2 28 23)(3 29 24)(4 30 19)(5 25 20)(6 26 21)(7 31 18)(8 32 13)(9 33 14)(10 34 15)(11 35 16)(12 36 17)
(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 31 4 34)(2 36 5 33)(3 35 6 32)(7 30 10 27)(8 29 11 26)(9 28 12 25)(13 24 16 21)(14 23 17 20)(15 22 18 19)
G:=sub<Sym(36)| (1,3,27)(2,23,19)(4,6,30)(5,20,22)(7,35,31)(8,13,12)(9,11,16)(10,32,34)(14,33,18)(15,17,36)(21,26,28)(24,29,25), (1,29,20)(2,30,21)(3,25,22)(4,26,23)(5,27,24)(6,28,19)(7,16,33)(8,17,34)(9,18,35)(10,13,36)(11,14,31)(12,15,32), (1,27,22)(2,28,23)(3,29,24)(4,30,19)(5,25,20)(6,26,21)(7,31,18)(8,32,13)(9,33,14)(10,34,15)(11,35,16)(12,36,17), (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), (1,31,4,34)(2,36,5,33)(3,35,6,32)(7,30,10,27)(8,29,11,26)(9,28,12,25)(13,24,16,21)(14,23,17,20)(15,22,18,19)>;
G:=Group( (1,3,27)(2,23,19)(4,6,30)(5,20,22)(7,35,31)(8,13,12)(9,11,16)(10,32,34)(14,33,18)(15,17,36)(21,26,28)(24,29,25), (1,29,20)(2,30,21)(3,25,22)(4,26,23)(5,27,24)(6,28,19)(7,16,33)(8,17,34)(9,18,35)(10,13,36)(11,14,31)(12,15,32), (1,27,22)(2,28,23)(3,29,24)(4,30,19)(5,25,20)(6,26,21)(7,31,18)(8,32,13)(9,33,14)(10,34,15)(11,35,16)(12,36,17), (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), (1,31,4,34)(2,36,5,33)(3,35,6,32)(7,30,10,27)(8,29,11,26)(9,28,12,25)(13,24,16,21)(14,23,17,20)(15,22,18,19) );
G=PermutationGroup([[(1,3,27),(2,23,19),(4,6,30),(5,20,22),(7,35,31),(8,13,12),(9,11,16),(10,32,34),(14,33,18),(15,17,36),(21,26,28),(24,29,25)], [(1,29,20),(2,30,21),(3,25,22),(4,26,23),(5,27,24),(6,28,19),(7,16,33),(8,17,34),(9,18,35),(10,13,36),(11,14,31),(12,15,32)], [(1,27,22),(2,28,23),(3,29,24),(4,30,19),(5,25,20),(6,26,21),(7,31,18),(8,32,13),(9,33,14),(10,34,15),(11,35,16),(12,36,17)], [(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)], [(1,31,4,34),(2,36,5,33),(3,35,6,32),(7,30,10,27),(8,29,11,26),(9,28,12,25),(13,24,16,21),(14,23,17,20),(15,22,18,19)]])
42 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | ··· | 3Q | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | ··· | 6Q | 12A | 12B | 12C | 12D |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | ··· | 6 | 27 | 27 | 1 | 1 | 2 | 2 | 2 | 6 | ··· | 6 | 27 | 27 | 27 | 27 |
42 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | 6 |
| type | + | + | + | + | - | - | |||||
| image | C1 | C2 | C4 | S3 | S3 | Dic3 | Dic3 | He3⋊C2 | He3⋊3C4 | He3⋊5S3 | He3⋊6Dic3 |
| kernel | He3⋊6Dic3 | C6×He3 | C3×He3 | C2×He3 | C32×C6 | He3 | C33 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 9 | 4 | 9 | 4 | 4 | 4 | 2 | 2 |
Matrix representation of He3⋊6Dic3 ►in GL5(𝔽13)
| 12 | 12 | 0 | 0 | 0 |
| 1 | 0 | 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 |
| 12 | 12 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 5 | 0 | 0 | 0 | 0 |
| 8 | 8 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(13))| [12,1,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[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],[12,1,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,3],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,3],[5,8,0,0,0,0,8,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;
He3⋊6Dic3 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_6{\rm Dic}_3 % in TeX
G:=Group("He3:6Dic3"); // GroupNames label
G:=SmallGroup(324,104);
// by ID
G=gap.SmallGroup(324,104);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,12,146,579,2164,382]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^6=1,e^2=d^3,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^-1,e*a*e^-1=a^-1,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations