direct product, metabelian, supersoluble, monomial
Aliases: C2×He3.2S3, He3.4D6, C9⋊S3⋊4C6, (C3×C18)⋊3C6, (C2×He3).8S3, C32.7(S3×C6), C6.8(C32⋊C6), He3⋊C3⋊3C22, (C2×C9⋊S3)⋊3C3, (C3×C9)⋊4(C2×C6), (C3×C6).18(C3×S3), C3.4(C2×C32⋊C6), (C2×He3⋊C3)⋊2C2, SmallGroup(324,73)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C9 — C2×He3.2S3 |
Generators and relations for C2×He3.2S3
G = < a,b,c,d,e,f | a2=b3=c3=d3=f2=1, e3=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd-1=bc-1, be=eb, fbf=b-1, cd=dc, ce=ec, fcf=c-1, ede-1=b-1cd, df=fd, fef=c-1e2 >
Subgroups: 448 in 56 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2, C3, C3, C22, S3, C6, C6, C9, C32, C32, D6, C2×C6, D9, C18, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C9, He3, He3, D18, S3×C6, C2×C3⋊S3, C32⋊C6, C9⋊S3, C3×C18, C2×He3, C2×He3, He3⋊C3, C2×C32⋊C6, C2×C9⋊S3, He3.2S3, C2×He3⋊C3, C2×He3.2S3
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, C3×S3, S3×C6, C32⋊C6, C2×C32⋊C6, He3.2S3, C2×He3.2S3
Character table of C2×He3.2S3
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 3F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 9A | 9B | 9C | 18A | 18B | 18C | |
| size | 1 | 1 | 27 | 27 | 2 | 6 | 9 | 9 | 18 | 18 | 2 | 6 | 9 | 9 | 18 | 18 | 27 | 27 | 27 | 27 | 6 | 6 | 6 | 6 | 6 | 6 | |
| 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 | trivial |
| ρ2 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ4 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | -1 | 1 | -1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | -1 | -1 | ζ65 | ζ6 | ζ65 | ζ6 | ζ32 | ζ65 | ζ3 | ζ6 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 6 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ6 | ζ65 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ8 | 1 | -1 | -1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | -1 | -1 | ζ65 | ζ6 | ζ65 | ζ6 | ζ6 | ζ3 | ζ65 | ζ32 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 6 |
| ρ9 | 1 | -1 | -1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | -1 | -1 | ζ6 | ζ65 | ζ6 | ζ65 | ζ65 | ζ32 | ζ6 | ζ3 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 6 |
| ρ10 | 1 | -1 | 1 | -1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | -1 | -1 | ζ6 | ζ65 | ζ6 | ζ65 | ζ3 | ζ6 | ζ32 | ζ65 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 6 |
| ρ11 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ12 | 1 | 1 | -1 | -1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ65 | ζ6 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ13 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 2 | -1 | -1 | -2 | -2 | -2 | -2 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ15 | 2 | 2 | 0 | 0 | 2 | 2 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | 2 | 2 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ16 | 2 | -2 | 0 | 0 | 2 | 2 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | -2 | -2 | 1+√-3 | 1-√-3 | ζ32 | ζ3 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | complex lifted from S3×C6 |
| ρ17 | 2 | 2 | 0 | 0 | 2 | 2 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | 2 | 2 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ18 | 2 | -2 | 0 | 0 | 2 | 2 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | -2 | -2 | 1-√-3 | 1+√-3 | ζ3 | ζ32 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | complex lifted from S3×C6 |
| ρ19 | 6 | 6 | 0 | 0 | 6 | -3 | 0 | 0 | 0 | 0 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C6 |
| ρ20 | 6 | -6 | 0 | 0 | 6 | -3 | 0 | 0 | 0 | 0 | -6 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×C32⋊C6 |
| ρ21 | 6 | -6 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ97-ζ94+2ζ92 | 2ζ95+ζ94+ζ92-ζ9 | orthogonal faithful |
| ρ22 | 6 | 6 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | orthogonal lifted from He3.2S3 |
| ρ23 | 6 | -6 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | ζ98+ζ97-ζ94+2ζ92 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ94-ζ92+2ζ9 | orthogonal faithful |
| ρ24 | 6 | 6 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | orthogonal lifted from He3.2S3 |
| ρ25 | 6 | 6 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | orthogonal lifted from He3.2S3 |
| ρ26 | 6 | -6 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ97-ζ94+2ζ92 | orthogonal faithful |
►On 54 points
Generators in S54
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 28)(8 29)(9 30)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)
(10 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 27)(37 43 40)(38 44 41)(39 45 42)(46 49 52)(47 50 53)(48 51 54)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)
(1 16 22)(2 11 23)(3 15 24)(4 10 25)(5 14 26)(6 18 27)(7 13 19)(8 17 20)(9 12 21)(28 40 46)(29 44 47)(30 39 48)(31 43 49)(32 38 50)(33 42 51)(34 37 52)(35 41 53)(36 45 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)
(2 9)(3 8)(4 7)(5 6)(10 13)(11 12)(14 18)(15 17)(19 25)(20 24)(21 23)(26 27)(28 34)(29 33)(30 32)(35 36)(37 40)(38 39)(41 45)(42 44)(46 52)(47 51)(48 50)(53 54)
G:=sub<Sym(54)| (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,28)(8,29)(9,30)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54), (10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27)(37,43,40)(38,44,41)(39,45,42)(46,49,52)(47,50,53)(48,51,54), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (1,16,22)(2,11,23)(3,15,24)(4,10,25)(5,14,26)(6,18,27)(7,13,19)(8,17,20)(9,12,21)(28,40,46)(29,44,47)(30,39,48)(31,43,49)(32,38,50)(33,42,51)(34,37,52)(35,41,53)(36,45,54), (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), (2,9)(3,8)(4,7)(5,6)(10,13)(11,12)(14,18)(15,17)(19,25)(20,24)(21,23)(26,27)(28,34)(29,33)(30,32)(35,36)(37,40)(38,39)(41,45)(42,44)(46,52)(47,51)(48,50)(53,54)>;
G:=Group( (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,28)(8,29)(9,30)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54), (10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27)(37,43,40)(38,44,41)(39,45,42)(46,49,52)(47,50,53)(48,51,54), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (1,16,22)(2,11,23)(3,15,24)(4,10,25)(5,14,26)(6,18,27)(7,13,19)(8,17,20)(9,12,21)(28,40,46)(29,44,47)(30,39,48)(31,43,49)(32,38,50)(33,42,51)(34,37,52)(35,41,53)(36,45,54), (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), (2,9)(3,8)(4,7)(5,6)(10,13)(11,12)(14,18)(15,17)(19,25)(20,24)(21,23)(26,27)(28,34)(29,33)(30,32)(35,36)(37,40)(38,39)(41,45)(42,44)(46,52)(47,51)(48,50)(53,54) );
G=PermutationGroup([[(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,28),(8,29),(9,30),(10,37),(11,38),(12,39),(13,40),(14,41),(15,42),(16,43),(17,44),(18,45),(19,46),(20,47),(21,48),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54)], [(10,16,13),(11,17,14),(12,18,15),(19,22,25),(20,23,26),(21,24,27),(37,43,40),(38,44,41),(39,45,42),(46,49,52),(47,50,53),(48,51,54)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54)], [(1,16,22),(2,11,23),(3,15,24),(4,10,25),(5,14,26),(6,18,27),(7,13,19),(8,17,20),(9,12,21),(28,40,46),(29,44,47),(30,39,48),(31,43,49),(32,38,50),(33,42,51),(34,37,52),(35,41,53),(36,45,54)], [(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)], [(2,9),(3,8),(4,7),(5,6),(10,13),(11,12),(14,18),(15,17),(19,25),(20,24),(21,23),(26,27),(28,34),(29,33),(30,32),(35,36),(37,40),(38,39),(41,45),(42,44),(46,52),(47,51),(48,50),(53,54)]])
Matrix representation of C2×He3.2S3 ►in GL6(𝔽19)
| 18 | 0 | 0 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 0 | 0 | 18 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 11 | 11 | 18 | 18 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 7 | 7 | 0 | 0 | 18 | 18 |
| 18 | 1 | 0 | 0 | 0 | 0 |
| 18 | 0 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 1 | 0 | 0 |
| 0 | 11 | 18 | 18 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 1 |
| 0 | 7 | 0 | 0 | 18 | 18 |
| 0 | 0 | 0 | 0 | 18 | 1 |
| 7 | 7 | 0 | 0 | 17 | 18 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 1 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 1 | 0 | 12 | 0 |
| 0 | 0 | 0 | 1 | 12 | 0 |
| 12 | 2 | 0 | 0 | 0 | 0 |
| 17 | 14 | 0 | 0 | 0 | 0 |
| 18 | 2 | 7 | 5 | 0 | 0 |
| 16 | 0 | 14 | 2 | 0 | 0 |
| 5 | 0 | 0 | 0 | 14 | 2 |
| 0 | 14 | 0 | 0 | 17 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 11 | 11 | 18 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 7 | 7 | 0 | 0 | 18 | 18 |
G:=sub<GL(6,GF(19))| [18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[1,0,11,0,0,7,0,1,11,0,0,7,0,0,18,1,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,1,18],[18,18,8,0,12,0,1,0,0,11,0,7,0,0,0,18,0,0,0,0,1,18,0,0,0,0,0,0,0,18,0,0,0,0,1,18],[0,7,0,1,0,0,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,18,17,8,8,12,12,1,18,0,0,0,0],[12,17,18,16,5,0,2,14,2,0,0,14,0,0,7,14,0,0,0,0,5,2,0,0,0,0,0,0,14,17,0,0,0,0,2,12],[0,1,0,11,0,7,1,0,0,11,0,7,0,0,1,18,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,18] >;
C2×He3.2S3 in GAP, Magma, Sage, TeX
C_2\times {\rm He}_3._2S_3 % in TeX
G:=Group("C2xHe3.2S3"); // GroupNames label
G:=SmallGroup(324,73);
// by ID
G=gap.SmallGroup(324,73);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,3171,303,453,2164,1096,7781]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^3=f^2=1,e^3=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,d*b*d^-1=b*c^-1,b*e=e*b,f*b*f=b^-1,c*d=d*c,c*e=e*c,f*c*f=c^-1,e*d*e^-1=b^-1*c*d,d*f=f*d,f*e*f=c^-1*e^2>;
// generators/relations
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