Aliases: He3.A4, C62.1C32, (C2×C6).5He3, C32.A4⋊2C3, C32.1(C3×A4), C3.6(C32⋊A4), C22⋊2(He3.C3), (C22×He3).1C3, (C3×C3.A4)⋊2C3, SmallGroup(324,53)
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=1, f3=b-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 >
Character table of He3.A4
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 9A | 9B | 9C | 9D | 9E | 9F | 9G | 9H | 9I | 9J | |
| size | 1 | 3 | 1 | 1 | 3 | 3 | 9 | 9 | 3 | 3 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 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 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | 1 | ζ32 | 1 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | 1 | ζ32 | 1 | ζ3 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | 1 | ζ3 | 1 | ζ32 | linear of order 3 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | 1 | ζ3 | 1 | linear of order 3 |
| ρ9 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ10 | 3 | -1 | 3 | 3 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ11 | 3 | 3 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 3 | 3 | 0 | 0 | 0 | 0 | -3+3√-3/2 | 0 | 0 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3 |
| ρ12 | 3 | -1 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | -1 | -1 | 2 | -1+√-3 | -1-√-3 | 2 | ζ6 | -1-√-3 | -1+√-3 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C32⋊A4 |
| ρ13 | 3 | -1 | 3 | 3 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | -1 | -1 | ζ65 | ζ6 | ζ6 | ζ6 | -1 | ζ65 | ζ65 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×A4 |
| ρ14 | 3 | -1 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | -1 | -1 | -1+√-3 | 2 | -1+√-3 | -1-√-3 | ζ6 | 2 | -1-√-3 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C32⋊A4 |
| ρ15 | 3 | 3 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 3 | 3 | 0 | 0 | 0 | 0 | -3-3√-3/2 | 0 | 0 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3 |
| ρ16 | 3 | -1 | 3 | 3 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | -1 | -1 | ζ6 | ζ65 | ζ65 | ζ65 | -1 | ζ6 | ζ6 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×A4 |
| ρ17 | 3 | -1 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | -1 | -1 | -1-√-3 | -1-√-3 | 2 | -1+√-3 | ζ6 | -1+√-3 | 2 | ζ65 | 0 | 0 | 0 | 0 | 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 | -1 | -1 | 2 | -1-√-3 | -1+√-3 | 2 | ζ65 | -1+√-3 | -1-√-3 | ζ6 | 0 | 0 | 0 | 0 | 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 | -1 | -1 | -1+√-3 | -1+√-3 | 2 | -1-√-3 | ζ65 | -1-√-3 | 2 | ζ6 | 0 | 0 | 0 | 0 | 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 | -1 | -1 | -1-√-3 | 2 | -1-√-3 | -1+√-3 | ζ65 | 2 | -1+√-3 | ζ6 | 0 | 0 | 0 | 0 | 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 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ97+2ζ94 | 2ζ97+ζ9 | ζ94+2ζ9 | 2ζ98+ζ95 | ζ98+2ζ92 | 2ζ95+ζ92 | 0 | 0 | 0 | 0 | complex lifted from He3.C3 |
| ρ22 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ94+2ζ9 | ζ97+2ζ94 | 2ζ97+ζ9 | ζ98+2ζ92 | 2ζ95+ζ92 | 2ζ98+ζ95 | 0 | 0 | 0 | 0 | complex lifted from He3.C3 |
| ρ23 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ95+ζ92 | ζ98+2ζ92 | 2ζ98+ζ95 | ζ94+2ζ9 | 2ζ97+ζ9 | ζ97+2ζ94 | 0 | 0 | 0 | 0 | complex lifted from He3.C3 |
| ρ24 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ98+ζ95 | 2ζ95+ζ92 | ζ98+2ζ92 | 2ζ97+ζ9 | ζ97+2ζ94 | ζ94+2ζ9 | 0 | 0 | 0 | 0 | complex lifted from He3.C3 |
| ρ25 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ98+2ζ92 | 2ζ98+ζ95 | 2ζ95+ζ92 | ζ97+2ζ94 | ζ94+2ζ9 | 2ζ97+ζ9 | 0 | 0 | 0 | 0 | complex lifted from He3.C3 |
| ρ26 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ97+ζ9 | ζ94+2ζ9 | ζ97+2ζ94 | 2ζ95+ζ92 | 2ζ98+ζ95 | ζ98+2ζ92 | 0 | 0 | 0 | 0 | complex lifted from He3.C3 |
| ρ27 | 9 | -3 | -9-9√-3/2 | -9+9√-3/2 | 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 | 0 | 0 | 0 | 0 | complex faithful |
| ρ28 | 9 | -3 | -9+9√-3/2 | -9-9√-3/2 | 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 | 0 | 0 | 0 | 0 | complex faithful |
►On 54 points
Generators in S54
(1 34 26)(2 35 27)(3 36 19)(4 28 20)(5 29 21)(6 30 22)(7 31 23)(8 32 24)(9 33 25)(10 45 52)(11 37 53)(12 38 54)(13 39 46)(14 40 47)(15 41 48)(16 42 49)(17 43 50)(18 44 51)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)
(2 27 29)(3 36 25)(5 21 32)(6 30 19)(8 24 35)(9 33 22)(10 45 49)(12 54 41)(13 39 52)(15 48 44)(16 42 46)(18 51 38)(20 23 26)(28 34 31)(37 43 40)(47 50 53)
(1 14)(2 15)(4 17)(5 18)(7 11)(8 12)(20 50)(21 51)(23 53)(24 54)(26 47)(27 48)(28 43)(29 44)(31 37)(32 38)(34 40)(35 41)
(2 15)(3 16)(5 18)(6 10)(8 12)(9 13)(19 49)(21 51)(22 52)(24 54)(25 46)(27 48)(29 44)(30 45)(32 38)(33 39)(35 41)(36 42)
(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)| (1,34,26)(2,35,27)(3,36,19)(4,28,20)(5,29,21)(6,30,22)(7,31,23)(8,32,24)(9,33,25)(10,45,52)(11,37,53)(12,38,54)(13,39,46)(14,40,47)(15,41,48)(16,42,49)(17,43,50)(18,44,51), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,52,49)(47,53,50)(48,54,51), (2,27,29)(3,36,25)(5,21,32)(6,30,19)(8,24,35)(9,33,22)(10,45,49)(12,54,41)(13,39,52)(15,48,44)(16,42,46)(18,51,38)(20,23,26)(28,34,31)(37,43,40)(47,50,53), (1,14)(2,15)(4,17)(5,18)(7,11)(8,12)(20,50)(21,51)(23,53)(24,54)(26,47)(27,48)(28,43)(29,44)(31,37)(32,38)(34,40)(35,41), (2,15)(3,16)(5,18)(6,10)(8,12)(9,13)(19,49)(21,51)(22,52)(24,54)(25,46)(27,48)(29,44)(30,45)(32,38)(33,39)(35,41)(36,42), (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( (1,34,26)(2,35,27)(3,36,19)(4,28,20)(5,29,21)(6,30,22)(7,31,23)(8,32,24)(9,33,25)(10,45,52)(11,37,53)(12,38,54)(13,39,46)(14,40,47)(15,41,48)(16,42,49)(17,43,50)(18,44,51), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,52,49)(47,53,50)(48,54,51), (2,27,29)(3,36,25)(5,21,32)(6,30,19)(8,24,35)(9,33,22)(10,45,49)(12,54,41)(13,39,52)(15,48,44)(16,42,46)(18,51,38)(20,23,26)(28,34,31)(37,43,40)(47,50,53), (1,14)(2,15)(4,17)(5,18)(7,11)(8,12)(20,50)(21,51)(23,53)(24,54)(26,47)(27,48)(28,43)(29,44)(31,37)(32,38)(34,40)(35,41), (2,15)(3,16)(5,18)(6,10)(8,12)(9,13)(19,49)(21,51)(22,52)(24,54)(25,46)(27,48)(29,44)(30,45)(32,38)(33,39)(35,41)(36,42), (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([[(1,34,26),(2,35,27),(3,36,19),(4,28,20),(5,29,21),(6,30,22),(7,31,23),(8,32,24),(9,33,25),(10,45,52),(11,37,53),(12,38,54),(13,39,46),(14,40,47),(15,41,48),(16,42,49),(17,43,50),(18,44,51)], [(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(28,34,31),(29,35,32),(30,36,33),(37,43,40),(38,44,41),(39,45,42),(46,52,49),(47,53,50),(48,54,51)], [(2,27,29),(3,36,25),(5,21,32),(6,30,19),(8,24,35),(9,33,22),(10,45,49),(12,54,41),(13,39,52),(15,48,44),(16,42,46),(18,51,38),(20,23,26),(28,34,31),(37,43,40),(47,50,53)], [(1,14),(2,15),(4,17),(5,18),(7,11),(8,12),(20,50),(21,51),(23,53),(24,54),(26,47),(27,48),(28,43),(29,44),(31,37),(32,38),(34,40),(35,41)], [(2,15),(3,16),(5,18),(6,10),(8,12),(9,13),(19,49),(21,51),(22,52),(24,54),(25,46),(27,48),(29,44),(30,45),(32,38),(33,39),(35,41),(36,42)], [(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)]])
Matrix representation of He3.A4 ►in GL6(𝔽19)
| 1 | 6 | 0 | 0 | 0 | 0 |
| 0 | 18 | 1 | 0 | 0 | 0 |
| 0 | 18 | 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 |
| 1 | 7 | 0 | 0 | 0 | 0 |
| 8 | 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 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 15 | 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 | 0 | 18 | 0 |
| 0 | 0 | 0 | 3 | 0 | 18 |
| 6 | 13 | 10 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 | 0 |
| 0 | 9 | 10 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 3 | 15 | 17 |
| 0 | 0 | 0 | 3 | 0 | 4 |
G:=sub<GL(6,GF(19))| [1,0,0,0,0,0,6,18,18,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,1,8,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,1,15,0,0,0,0,0,18],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,3,0,0,0,0,18,0,0,0,0,0,0,18],[6,0,0,0,0,0,13,0,9,0,0,0,10,10,10,0,0,0,0,0,0,0,3,3,0,0,0,1,15,0,0,0,0,0,17,4] >;
He3.A4 in GAP, Magma, Sage, TeX
{\rm He}_3.A_4 % in TeX
G:=Group("He3.A4"); // GroupNames label
G:=SmallGroup(324,53);
// by ID
G=gap.SmallGroup(324,53);
# by ID
G:=PCGroup([6,-3,-3,-3,-3,-2,2,162,145,386,4864,8753]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^2=e^2=1,f^3=b^-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.A4 in TeX
Character table of He3.A4 in TeX