Aliases: He3⋊1A4, C62.2C32, C32⋊A4⋊2C3, (C2×C6).6He3, C32.2(C3×A4), C3.7(C32⋊A4), (C22×He3)⋊1C3, C22⋊2(He3⋊C3), (C3×C3.A4)⋊3C3, SmallGroup(324,54)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for He3⋊A4
G = < a,b,c,d,e,f | a3=b3=c3=d2=e2=f3=1, ab=ba, cac-1=faf-1=ab-1, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf-1=a-1b-1c, fdf-1=de=ed, fef-1=d >
Character table of He3⋊A4
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 3I | 3J | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 9A | 9B | 9C | 9D | 9E | 9F | |
| size | 1 | 3 | 1 | 1 | 3 | 3 | 9 | 9 | 36 | 36 | 36 | 36 | 3 | 3 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | |
| 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 | ζ3 | ζ32 | 1 | ζ3 | 1 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | 1 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ9 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ10 | 3 | -1 | 3 | 3 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ11 | 3 | -1 | 3 | 3 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | -1 | -1 | ζ6 | ζ65 | ζ65 | ζ65 | -1 | ζ6 | ζ6 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×A4 |
| ρ12 | 3 | -1 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1-√-3 | 2 | -1+√-3 | -1-√-3 | ζ6 | -1+√-3 | 2 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C32⋊A4 |
| ρ13 | 3 | -1 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | -1+√-3 | 2 | -1-√-3 | ζ65 | -1+√-3 | -1-√-3 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C32⋊A4 |
| ρ14 | 3 | 3 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 | 0 | 0 | 0 | 0 | -3-3√-3/2 | 0 | 0 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3 |
| ρ15 | 3 | -1 | 3 | 3 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | -1 | -1 | ζ65 | ζ6 | ζ6 | ζ6 | -1 | ζ65 | ζ65 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×A4 |
| ρ16 | 3 | 3 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 | 0 | 0 | 0 | 0 | -3+3√-3/2 | 0 | 0 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3 |
| ρ17 | 3 | -1 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1+√-3 | -1+√-3 | -1-√-3 | 2 | ζ6 | 2 | -1-√-3 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C32⋊A4 |
| ρ18 | 3 | -1 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1+√-3 | 2 | -1-√-3 | -1+√-3 | ζ65 | -1-√-3 | 2 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C32⋊A4 |
| ρ19 | 3 | -1 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | -1-√-3 | 2 | -1+√-3 | ζ6 | -1-√-3 | -1+√-3 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C32⋊A4 |
| ρ20 | 3 | -1 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1-√-3 | -1-√-3 | -1+√-3 | 2 | ζ65 | 2 | -1+√-3 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C32⋊A4 |
| ρ21 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ98+2ζ95 | ζ95+2ζ92 | 2ζ97+ζ94 | 2ζ94+ζ9 | ζ97+2ζ9 | 2ζ98+ζ92 | complex lifted from He3⋊C3 |
| ρ22 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ97+2ζ9 | 2ζ94+ζ9 | ζ98+2ζ95 | 2ζ98+ζ92 | ζ95+2ζ92 | 2ζ97+ζ94 | complex lifted from He3⋊C3 |
| ρ23 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ97+ζ94 | ζ97+2ζ9 | 2ζ98+ζ92 | ζ95+2ζ92 | ζ98+2ζ95 | 2ζ94+ζ9 | complex lifted from He3⋊C3 |
| ρ24 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ94+ζ9 | 2ζ97+ζ94 | ζ95+2ζ92 | ζ98+2ζ95 | 2ζ98+ζ92 | ζ97+2ζ9 | complex lifted from He3⋊C3 |
| ρ25 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ95+2ζ92 | 2ζ98+ζ92 | ζ97+2ζ9 | 2ζ97+ζ94 | 2ζ94+ζ9 | ζ98+2ζ95 | complex lifted from He3⋊C3 |
| ρ26 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ98+ζ92 | ζ98+2ζ95 | 2ζ94+ζ9 | ζ97+2ζ9 | 2ζ97+ζ94 | ζ95+2ζ92 | complex lifted from He3⋊C3 |
| ρ27 | 9 | -3 | -9-9√-3/2 | -9+9√-3/2 | 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 | 0 | 0 | complex faithful |
| ρ28 | 9 | -3 | -9+9√-3/2 | -9-9√-3/2 | 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 | 0 | 0 | complex faithful |
►On 54 points
Generators in S54
(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)
(1 10 22)(2 11 23)(3 12 24)(4 16 26)(5 17 27)(6 18 25)(7 43 49)(8 44 50)(9 45 51)(13 53 41)(14 54 42)(15 52 40)(19 36 48)(20 34 46)(21 35 47)(28 38 32)(29 39 33)(30 37 31)
(2 11 23)(3 24 12)(4 26 16)(6 18 25)(7 45 44)(8 49 9)(13 52 54)(14 41 15)(19 34 21)(20 47 48)(28 39 30)(29 31 32)(33 37 38)(35 36 46)(40 42 53)(43 51 50)
(1 17)(2 18)(3 16)(4 24)(5 22)(6 23)(7 53)(8 54)(9 52)(10 27)(11 25)(12 26)(13 49)(14 50)(15 51)(40 45)(41 43)(42 44)
(7 53)(8 54)(9 52)(13 49)(14 50)(15 51)(19 38)(20 39)(21 37)(28 48)(29 46)(30 47)(31 35)(32 36)(33 34)(40 45)(41 43)(42 44)
(1 9 29)(2 43 31)(3 50 38)(4 42 48)(5 15 34)(6 53 21)(7 37 23)(8 32 12)(10 45 39)(11 49 30)(13 47 25)(14 19 16)(17 52 46)(18 41 35)(20 27 40)(22 51 33)(24 44 28)(26 54 36)
G:=sub<Sym(54)| (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), (1,10,22)(2,11,23)(3,12,24)(4,16,26)(5,17,27)(6,18,25)(7,43,49)(8,44,50)(9,45,51)(13,53,41)(14,54,42)(15,52,40)(19,36,48)(20,34,46)(21,35,47)(28,38,32)(29,39,33)(30,37,31), (2,11,23)(3,24,12)(4,26,16)(6,18,25)(7,45,44)(8,49,9)(13,52,54)(14,41,15)(19,34,21)(20,47,48)(28,39,30)(29,31,32)(33,37,38)(35,36,46)(40,42,53)(43,51,50), (1,17)(2,18)(3,16)(4,24)(5,22)(6,23)(7,53)(8,54)(9,52)(10,27)(11,25)(12,26)(13,49)(14,50)(15,51)(40,45)(41,43)(42,44), (7,53)(8,54)(9,52)(13,49)(14,50)(15,51)(19,38)(20,39)(21,37)(28,48)(29,46)(30,47)(31,35)(32,36)(33,34)(40,45)(41,43)(42,44), (1,9,29)(2,43,31)(3,50,38)(4,42,48)(5,15,34)(6,53,21)(7,37,23)(8,32,12)(10,45,39)(11,49,30)(13,47,25)(14,19,16)(17,52,46)(18,41,35)(20,27,40)(22,51,33)(24,44,28)(26,54,36)>;
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)(37,38,39)(40,41,42)(43,44,45)(46,47,48)(49,50,51)(52,53,54), (1,10,22)(2,11,23)(3,12,24)(4,16,26)(5,17,27)(6,18,25)(7,43,49)(8,44,50)(9,45,51)(13,53,41)(14,54,42)(15,52,40)(19,36,48)(20,34,46)(21,35,47)(28,38,32)(29,39,33)(30,37,31), (2,11,23)(3,24,12)(4,26,16)(6,18,25)(7,45,44)(8,49,9)(13,52,54)(14,41,15)(19,34,21)(20,47,48)(28,39,30)(29,31,32)(33,37,38)(35,36,46)(40,42,53)(43,51,50), (1,17)(2,18)(3,16)(4,24)(5,22)(6,23)(7,53)(8,54)(9,52)(10,27)(11,25)(12,26)(13,49)(14,50)(15,51)(40,45)(41,43)(42,44), (7,53)(8,54)(9,52)(13,49)(14,50)(15,51)(19,38)(20,39)(21,37)(28,48)(29,46)(30,47)(31,35)(32,36)(33,34)(40,45)(41,43)(42,44), (1,9,29)(2,43,31)(3,50,38)(4,42,48)(5,15,34)(6,53,21)(7,37,23)(8,32,12)(10,45,39)(11,49,30)(13,47,25)(14,19,16)(17,52,46)(18,41,35)(20,27,40)(22,51,33)(24,44,28)(26,54,36) );
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),(37,38,39),(40,41,42),(43,44,45),(46,47,48),(49,50,51),(52,53,54)], [(1,10,22),(2,11,23),(3,12,24),(4,16,26),(5,17,27),(6,18,25),(7,43,49),(8,44,50),(9,45,51),(13,53,41),(14,54,42),(15,52,40),(19,36,48),(20,34,46),(21,35,47),(28,38,32),(29,39,33),(30,37,31)], [(2,11,23),(3,24,12),(4,26,16),(6,18,25),(7,45,44),(8,49,9),(13,52,54),(14,41,15),(19,34,21),(20,47,48),(28,39,30),(29,31,32),(33,37,38),(35,36,46),(40,42,53),(43,51,50)], [(1,17),(2,18),(3,16),(4,24),(5,22),(6,23),(7,53),(8,54),(9,52),(10,27),(11,25),(12,26),(13,49),(14,50),(15,51),(40,45),(41,43),(42,44)], [(7,53),(8,54),(9,52),(13,49),(14,50),(15,51),(19,38),(20,39),(21,37),(28,48),(29,46),(30,47),(31,35),(32,36),(33,34),(40,45),(41,43),(42,44)], [(1,9,29),(2,43,31),(3,50,38),(4,42,48),(5,15,34),(6,53,21),(7,37,23),(8,32,12),(10,45,39),(11,49,30),(13,47,25),(14,19,16),(17,52,46),(18,41,35),(20,27,40),(22,51,33),(24,44,28),(26,54,36)]])
Matrix representation of He3⋊A4 ►in GL6(𝔽19)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 7 | 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 |
| 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 | 14 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 18 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 5 | 18 | 0 |
| 0 | 0 | 0 | 9 | 0 | 18 |
| 13 | 15 | 15 | 0 | 0 | 0 |
| 10 | 15 | 10 | 0 | 0 | 0 |
| 13 | 13 | 10 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 17 | 0 |
| 0 | 0 | 0 | 8 | 14 | 1 |
| 0 | 0 | 0 | 3 | 10 | 0 |
G:=sub<GL(6,GF(19))| [0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,14,0,0,0,0,0,1,0,0,0,0,0,0,18],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,5,9,0,0,0,0,18,0,0,0,0,0,0,18],[13,10,13,0,0,0,15,15,13,0,0,0,15,10,10,0,0,0,0,0,0,5,8,3,0,0,0,17,14,10,0,0,0,0,1,0] >;
He3⋊A4 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes A_4 % in TeX
G:=Group("He3:A4"); // GroupNames label
G:=SmallGroup(324,54);
// by ID
G=gap.SmallGroup(324,54);
# by ID
G:=PCGroup([6,-3,-3,-3,-3,-2,2,145,650,224,4864,8753]);
// 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=f*a*f^-1=a*b^-1,a*d=d*a,a*e=e*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^-1*c,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations
Export
Subgroup lattice of He3⋊A4 in TeX
Character table of He3⋊A4 in TeX