Aliases: He3:2A4, C62.3C32, C22:1C3wrC3, (C2xC6).7He3, (C32xA4):1C3, C32.A4:3C3, C32.3(C3xA4), C3.8(C32:A4), (C22xHe3):2C3, SmallGroup(324,55)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for He3:2A4
G = < a,b,c,d,e,f | a3=b3=c3=d2=e2=f3=1, ab=ba, cac-1=ab-1, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf-1=a-1bc, fdf-1=de=ed, fef-1=d >
Quotients: C1, C3, C32, A4, He3, C3xA4, C3wrC3, C32:A4, He3:2A4
3C2
3C3
9C3
12C3
12C3
12C3
3C6
9C6
9C6
9C6
9C6
3C32
12C9
12C32
12C32
12C9
12C32
12C32
3A4
3A4
3A4
4C33
3C62
Character table of He3:2A4
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 3I | 3J | 3K | 3L | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 9A | 9B | 9C | 9D | |
| size | 1 | 3 | 1 | 1 | 3 | 3 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 3 | 3 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 36 | 36 | 36 | 36 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | 1 | 1 | linear of order 3 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ32 | 1 | ζ3 | 1 | 1 | ζ3 | ζ32 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | 1 | 1 | linear of order 3 |
| ρ9 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ3 | 1 | ζ32 | 1 | 1 | ζ32 | ζ3 | linear of order 3 |
| ρ10 | 3 | -1 | 3 | 3 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ11 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 3+√-3/2 | -3-√-3/2 | 3-√-3/2 | √-3 | -3+√-3/2 | -√-3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3wrC3 |
| ρ12 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 3-√-3/2 | -3+√-3/2 | 3+√-3/2 | -√-3 | -3-√-3/2 | √-3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3wrC3 |
| ρ13 | 3 | -1 | 3 | 3 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | ζ65 | ζ65 | -1 | ζ6 | ζ6 | ζ6 | -1 | ζ65 | 0 | 0 | 0 | 0 | complex lifted from C3xA4 |
| ρ14 | 3 | -1 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1+√-3 | -1-√-3 | ζ6 | 2 | -1-√-3 | -1+√-3 | ζ65 | 2 | 0 | 0 | 0 | 0 | complex lifted from C32:A4 |
| ρ15 | 3 | 3 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 | 0 | 0 | -3-3√-3/2 | 0 | 0 | 0 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3 |
| ρ16 | 3 | -1 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | -1-√-3 | ζ65 | -1-√-3 | 2 | -1+√-3 | ζ6 | -1+√-3 | 0 | 0 | 0 | 0 | complex lifted from C32:A4 |
| ρ17 | 3 | -1 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1-√-3 | 2 | ζ6 | -1-√-3 | -1+√-3 | 2 | ζ65 | -1+√-3 | 0 | 0 | 0 | 0 | complex lifted from C32:A4 |
| ρ18 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | √-3 | 3+√-3/2 | -√-3 | -3+√-3/2 | 3-√-3/2 | -3-√-3/2 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3wrC3 |
| ρ19 | 3 | -1 | 3 | 3 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | ζ6 | ζ6 | -1 | ζ65 | ζ65 | ζ65 | -1 | ζ6 | 0 | 0 | 0 | 0 | complex lifted from C3xA4 |
| ρ20 | 3 | -1 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1-√-3 | -1+√-3 | ζ65 | 2 | -1+√-3 | -1-√-3 | ζ6 | 2 | 0 | 0 | 0 | 0 | complex lifted from C32:A4 |
| ρ21 | 3 | -1 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1+√-3 | 2 | ζ65 | -1+√-3 | -1-√-3 | 2 | ζ6 | -1-√-3 | 0 | 0 | 0 | 0 | complex lifted from C32:A4 |
| ρ22 | 3 | 3 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 | 0 | 0 | -3+3√-3/2 | 0 | 0 | 0 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3 |
| ρ23 | 3 | -1 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | -1+√-3 | ζ6 | -1+√-3 | 2 | -1-√-3 | ζ65 | -1-√-3 | 0 | 0 | 0 | 0 | complex lifted from C32:A4 |
| ρ24 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | -3-√-3/2 | -√-3 | -3+√-3/2 | 3+√-3/2 | √-3 | 3-√-3/2 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3wrC3 |
| ρ25 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | -3+√-3/2 | √-3 | -3-√-3/2 | 3-√-3/2 | -√-3 | 3+√-3/2 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3wrC3 |
| ρ26 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | -√-3 | 3-√-3/2 | √-3 | -3-√-3/2 | 3+√-3/2 | -3+√-3/2 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3wrC3 |
| ρ27 | 9 | -3 | -9-9√-3/2 | -9+9√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3+3√-3/2 | 3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ28 | 9 | -3 | -9+9√-3/2 | -9-9√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3-3√-3/2 | 3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 36 points
Generators in S36
(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 9 5)(2 10 6)(3 11 7)(4 12 8)(13 15 14)(16 17 18)(19 20 21)(22 24 23)(25 26 27)(28 30 29)(31 32 33)(34 36 35)
(1 24 27)(2 35 17)(3 28 20)(4 13 31)(5 22 26)(6 36 16)(7 29 19)(8 14 33)(9 23 25)(10 34 18)(11 30 21)(12 15 32)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 35)(14 36)(15 34)(16 33)(17 31)(18 32)(19 26)(20 27)(21 25)(22 29)(23 30)(24 28)
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 28)(14 29)(15 30)(16 26)(17 27)(18 25)(19 33)(20 31)(21 32)(22 36)(23 34)(24 35)
(1 5 9)(2 8 11)(3 6 12)(4 7 10)(13 30 36)(14 28 34)(15 29 35)(16 32 20)(17 33 21)(18 31 19)(22 24 23)(25 27 26)
G:=sub<Sym(36)| (13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,15,14)(16,17,18)(19,20,21)(22,24,23)(25,26,27)(28,30,29)(31,32,33)(34,36,35), (1,24,27)(2,35,17)(3,28,20)(4,13,31)(5,22,26)(6,36,16)(7,29,19)(8,14,33)(9,23,25)(10,34,18)(11,30,21)(12,15,32), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,35)(14,36)(15,34)(16,33)(17,31)(18,32)(19,26)(20,27)(21,25)(22,29)(23,30)(24,28), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,28)(14,29)(15,30)(16,26)(17,27)(18,25)(19,33)(20,31)(21,32)(22,36)(23,34)(24,35), (1,5,9)(2,8,11)(3,6,12)(4,7,10)(13,30,36)(14,28,34)(15,29,35)(16,32,20)(17,33,21)(18,31,19)(22,24,23)(25,27,26)>;
G:=Group( (13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,15,14)(16,17,18)(19,20,21)(22,24,23)(25,26,27)(28,30,29)(31,32,33)(34,36,35), (1,24,27)(2,35,17)(3,28,20)(4,13,31)(5,22,26)(6,36,16)(7,29,19)(8,14,33)(9,23,25)(10,34,18)(11,30,21)(12,15,32), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,35)(14,36)(15,34)(16,33)(17,31)(18,32)(19,26)(20,27)(21,25)(22,29)(23,30)(24,28), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,28)(14,29)(15,30)(16,26)(17,27)(18,25)(19,33)(20,31)(21,32)(22,36)(23,34)(24,35), (1,5,9)(2,8,11)(3,6,12)(4,7,10)(13,30,36)(14,28,34)(15,29,35)(16,32,20)(17,33,21)(18,31,19)(22,24,23)(25,27,26) );
G=PermutationGroup([[(13,14,15),(16,17,18),(19,20,21),(22,23,24),(25,26,27),(28,29,30),(31,32,33),(34,35,36)], [(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,15,14),(16,17,18),(19,20,21),(22,24,23),(25,26,27),(28,30,29),(31,32,33),(34,36,35)], [(1,24,27),(2,35,17),(3,28,20),(4,13,31),(5,22,26),(6,36,16),(7,29,19),(8,14,33),(9,23,25),(10,34,18),(11,30,21),(12,15,32)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,35),(14,36),(15,34),(16,33),(17,31),(18,32),(19,26),(20,27),(21,25),(22,29),(23,30),(24,28)], [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,28),(14,29),(15,30),(16,26),(17,27),(18,25),(19,33),(20,31),(21,32),(22,36),(23,34),(24,35)], [(1,5,9),(2,8,11),(3,6,12),(4,7,10),(13,30,36),(14,28,34),(15,29,35),(16,32,20),(17,33,21),(18,31,19),(22,24,23),(25,27,26)]])
Matrix representation of He3:2A4 ►in GL6(F19)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 1 |
| 0 | 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 1 | 18 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 0 | 18 | 0 | 1 |
| 0 | 0 | 0 | 18 | 1 | 0 |
| 7 | 0 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(19))| [1,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,18,18,18,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,18,18,0,0,0,0,0,1,0,0,0,0,1,0],[7,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0] >;
He3:2A4 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_2A_4 % in TeX
G:=Group("He3:2A4"); // GroupNames label
G:=SmallGroup(324,55);
// by ID
G=gap.SmallGroup(324,55);
# by ID
G:=PCGroup([6,-3,-3,-3,-3,-2,2,145,224,4864,8753]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^2=e^2=f^3=1,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,f*c*f^-1=a^-1*b*c,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations
Export
Subgroup lattice of He3:2A4 in TeX
Character table of He3:2A4 in TeX