non-abelian, supersoluble, monomial
Aliases: (C32×C9).16S3, C3.He3⋊1S3, C33.40(C3⋊S3), C3.6(He3⋊5S3), C3⋊(3- 1+2.S3), C32.4(C33⋊C2), (C3×3- 1+2).9S3, C32.30(He3⋊C2), 3- 1+2.1(C3⋊S3), (C3×C9).8(C3⋊S3), (C3×C3.He3)⋊2C2, SmallGroup(486,188)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C3.He3 — (C32×C9).S3 |
Generators and relations for (C32×C9).S3
G = < a,b,c,d,e | a3=b3=c9=e2=1, d3=c3, ab=ba, ac=ca, dad-1=ac3, eae=abc6, bc=cb, bd=db, ebe=b-1, dcd-1=a-1bc, ece=c-1, ede=c6d2 >
Subgroups: 790 in 114 conjugacy classes, 35 normal (8 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, C3×C9, 3- 1+2, 3- 1+2, C33, C3×D9, C9⋊C6, C9⋊S3, C3×C3⋊S3, C3.He3, C32×C9, C3×3- 1+2, 3- 1+2.S3, C3×C9⋊S3, C33.S3, C3×C3.He3, (C32×C9).S3
Quotients: C1, C2, S3, C3⋊S3, He3⋊C2, C33⋊C2, 3- 1+2.S3, He3⋊5S3, (C32×C9).S3
Character table of (C32×C9).S3
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 6A | 6B | 9A | 9B | 9C | 9D | 9E | 9F | 9G | 9H | 9I | 9J | 9K | 9L | 9M | 9N | 9O | 9P | 9Q | 9R | |
| size | 1 | 81 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 81 | 81 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | |
| 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 | 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 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ4 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ5 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | 2 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ9 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ13 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | orthogonal lifted from S3 |
| ρ14 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ15 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | orthogonal lifted from S3 |
| ρ16 | 3 | 1 | 3 | 3 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | -3+3√-3/2 | -3-3√-3/2 | ζ3 | ζ32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3⋊C2 |
| ρ17 | 3 | -1 | 3 | 3 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | -3+3√-3/2 | -3-3√-3/2 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3⋊C2 |
| ρ18 | 3 | -1 | 3 | 3 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | -3-3√-3/2 | -3+3√-3/2 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3⋊C2 |
| ρ19 | 3 | 1 | 3 | 3 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | -3-3√-3/2 | -3+3√-3/2 | ζ32 | ζ3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3⋊C2 |
| ρ20 | 6 | 0 | -3 | 6 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 3- 1+2.S3 |
| ρ21 | 6 | 0 | -3 | -3 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ98-ζ94+ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 3- 1+2.S3 |
| ρ22 | 6 | 0 | -3 | -3 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ95+2ζ94-ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 3- 1+2.S3 |
| ρ23 | 6 | 0 | -3 | 6 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 3- 1+2.S3 |
| ρ24 | 6 | 0 | -3 | -3 | -3 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 3- 1+2.S3 |
| ρ25 | 6 | 0 | -3 | -3 | -3 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 3- 1+2.S3 |
| ρ26 | 6 | 0 | -3 | 6 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 3- 1+2.S3 |
| ρ27 | 6 | 0 | -3 | -3 | -3 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 3- 1+2.S3 |
| ρ28 | 6 | 0 | -3 | -3 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ98+2ζ97+ζ94+ζ92 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 3- 1+2.S3 |
| ρ29 | 6 | 0 | 6 | -3 | -3 | -3 | -3-3√-3 | -3+3√-3 | 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 | 0 | 0 | complex lifted from He3⋊5S3 |
| ρ30 | 6 | 0 | 6 | -3 | -3 | -3 | -3+3√-3 | -3-3√-3 | 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 | 0 | 0 | complex lifted from He3⋊5S3 |
►On 81 points
Generators in S81
(1 28 61)(2 29 62)(3 30 63)(4 31 55)(5 32 56)(6 33 57)(7 34 58)(8 35 59)(9 36 60)(10 37 64)(11 38 65)(12 39 66)(13 40 67)(14 41 68)(15 42 69)(16 43 70)(17 44 71)(18 45 72)(19 49 79)(20 50 80)(21 51 81)(22 52 73)(23 53 74)(24 54 75)(25 46 76)(26 47 77)(27 48 78)
(1 31 58)(2 32 59)(3 33 60)(4 34 61)(5 35 62)(6 36 63)(7 28 55)(8 29 56)(9 30 57)(10 37 64)(11 38 65)(12 39 66)(13 40 67)(14 41 68)(15 42 69)(16 43 70)(17 44 71)(18 45 72)(19 46 73)(20 47 74)(21 48 75)(22 49 76)(23 50 77)(24 51 78)(25 52 79)(26 53 80)(27 54 81)
(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)(55 56 57 58 59 60 61 62 63)(64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81)
(1 10 19 4 13 22 7 16 25)(2 17 26 5 11 20 8 14 23)(3 15 24 6 18 27 9 12 21)(28 43 52 31 37 46 34 40 49)(29 41 50 32 44 53 35 38 47)(30 39 48 33 42 51 36 45 54)(55 70 79 58 64 73 61 67 76)(56 68 77 59 71 80 62 65 74)(57 66 75 60 69 78 63 72 81)
(2 9)(3 8)(4 7)(5 6)(10 25)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)(17 27)(18 26)(28 61)(29 60)(30 59)(31 58)(32 57)(33 56)(34 55)(35 63)(36 62)(37 79)(38 78)(39 77)(40 76)(41 75)(42 74)(43 73)(44 81)(45 80)(46 70)(47 69)(48 68)(49 67)(50 66)(51 65)(52 64)(53 72)(54 71)
G:=sub<Sym(81)| (1,28,61)(2,29,62)(3,30,63)(4,31,55)(5,32,56)(6,33,57)(7,34,58)(8,35,59)(9,36,60)(10,37,64)(11,38,65)(12,39,66)(13,40,67)(14,41,68)(15,42,69)(16,43,70)(17,44,71)(18,45,72)(19,49,79)(20,50,80)(21,51,81)(22,52,73)(23,53,74)(24,54,75)(25,46,76)(26,47,77)(27,48,78), (1,31,58)(2,32,59)(3,33,60)(4,34,61)(5,35,62)(6,36,63)(7,28,55)(8,29,56)(9,30,57)(10,37,64)(11,38,65)(12,39,66)(13,40,67)(14,41,68)(15,42,69)(16,43,70)(17,44,71)(18,45,72)(19,46,73)(20,47,74)(21,48,75)(22,49,76)(23,50,77)(24,51,78)(25,52,79)(26,53,80)(27,54,81), (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)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81), (1,10,19,4,13,22,7,16,25)(2,17,26,5,11,20,8,14,23)(3,15,24,6,18,27,9,12,21)(28,43,52,31,37,46,34,40,49)(29,41,50,32,44,53,35,38,47)(30,39,48,33,42,51,36,45,54)(55,70,79,58,64,73,61,67,76)(56,68,77,59,71,80,62,65,74)(57,66,75,60,69,78,63,72,81), (2,9)(3,8)(4,7)(5,6)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,27)(18,26)(28,61)(29,60)(30,59)(31,58)(32,57)(33,56)(34,55)(35,63)(36,62)(37,79)(38,78)(39,77)(40,76)(41,75)(42,74)(43,73)(44,81)(45,80)(46,70)(47,69)(48,68)(49,67)(50,66)(51,65)(52,64)(53,72)(54,71)>;
G:=Group( (1,28,61)(2,29,62)(3,30,63)(4,31,55)(5,32,56)(6,33,57)(7,34,58)(8,35,59)(9,36,60)(10,37,64)(11,38,65)(12,39,66)(13,40,67)(14,41,68)(15,42,69)(16,43,70)(17,44,71)(18,45,72)(19,49,79)(20,50,80)(21,51,81)(22,52,73)(23,53,74)(24,54,75)(25,46,76)(26,47,77)(27,48,78), (1,31,58)(2,32,59)(3,33,60)(4,34,61)(5,35,62)(6,36,63)(7,28,55)(8,29,56)(9,30,57)(10,37,64)(11,38,65)(12,39,66)(13,40,67)(14,41,68)(15,42,69)(16,43,70)(17,44,71)(18,45,72)(19,46,73)(20,47,74)(21,48,75)(22,49,76)(23,50,77)(24,51,78)(25,52,79)(26,53,80)(27,54,81), (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)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81), (1,10,19,4,13,22,7,16,25)(2,17,26,5,11,20,8,14,23)(3,15,24,6,18,27,9,12,21)(28,43,52,31,37,46,34,40,49)(29,41,50,32,44,53,35,38,47)(30,39,48,33,42,51,36,45,54)(55,70,79,58,64,73,61,67,76)(56,68,77,59,71,80,62,65,74)(57,66,75,60,69,78,63,72,81), (2,9)(3,8)(4,7)(5,6)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,27)(18,26)(28,61)(29,60)(30,59)(31,58)(32,57)(33,56)(34,55)(35,63)(36,62)(37,79)(38,78)(39,77)(40,76)(41,75)(42,74)(43,73)(44,81)(45,80)(46,70)(47,69)(48,68)(49,67)(50,66)(51,65)(52,64)(53,72)(54,71) );
G=PermutationGroup([[(1,28,61),(2,29,62),(3,30,63),(4,31,55),(5,32,56),(6,33,57),(7,34,58),(8,35,59),(9,36,60),(10,37,64),(11,38,65),(12,39,66),(13,40,67),(14,41,68),(15,42,69),(16,43,70),(17,44,71),(18,45,72),(19,49,79),(20,50,80),(21,51,81),(22,52,73),(23,53,74),(24,54,75),(25,46,76),(26,47,77),(27,48,78)], [(1,31,58),(2,32,59),(3,33,60),(4,34,61),(5,35,62),(6,36,63),(7,28,55),(8,29,56),(9,30,57),(10,37,64),(11,38,65),(12,39,66),(13,40,67),(14,41,68),(15,42,69),(16,43,70),(17,44,71),(18,45,72),(19,46,73),(20,47,74),(21,48,75),(22,49,76),(23,50,77),(24,51,78),(25,52,79),(26,53,80),(27,54,81)], [(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),(55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81)], [(1,10,19,4,13,22,7,16,25),(2,17,26,5,11,20,8,14,23),(3,15,24,6,18,27,9,12,21),(28,43,52,31,37,46,34,40,49),(29,41,50,32,44,53,35,38,47),(30,39,48,33,42,51,36,45,54),(55,70,79,58,64,73,61,67,76),(56,68,77,59,71,80,62,65,74),(57,66,75,60,69,78,63,72,81)], [(2,9),(3,8),(4,7),(5,6),(10,25),(11,24),(12,23),(13,22),(14,21),(15,20),(16,19),(17,27),(18,26),(28,61),(29,60),(30,59),(31,58),(32,57),(33,56),(34,55),(35,63),(36,62),(37,79),(38,78),(39,77),(40,76),(41,75),(42,74),(43,73),(44,81),(45,80),(46,70),(47,69),(48,68),(49,67),(50,66),(51,65),(52,64),(53,72),(54,71)]])
Matrix representation of (C32×C9).S3 ►in GL8(𝔽19)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 18 | 18 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| 0 | 0 | 18 | 0 | 18 | 18 | 18 | 18 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 18 | 18 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 18 | 18 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 7 | 0 | 0 |
| 0 | 0 | 0 | 7 | 5 | 0 | 7 | 5 |
| 0 | 0 | 14 | 2 | 0 | 14 | 14 | 2 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 18 | 18 | 18 | 18 | 17 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 18 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 18 |
| 0 | 0 | 18 | 18 | 18 | 18 | 18 | 17 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(8,GF(19))| [0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,1,0,0,0,18,0,0,18,18,0,0,1,0,0,0,0,0,1,0,0,18,0,0,0,0,0,1,0,18,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18],[0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[18,1,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,5,7,0,0,0,14,0,0,12,17,0,0,7,2,0,0,0,0,2,14,5,0,0,0,0,0,5,7,0,14,0,0,0,0,0,0,7,14,0,0,0,0,0,0,5,2],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,0,1,0,0,0,0,0,18,0,0,1,0,0,0,0,18,0,0,0,0,1,0,0,18,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,17,18,0,0,1,1],[1,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,1,0,18,0,0,0,0,1,0,0,18,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,0,18,17,1,1] >;
(C32×C9).S3 in GAP, Magma, Sage, TeX
(C_3^2\times C_9).S_3
% in TeX
G:=Group("(C3^2xC9).S3"); // GroupNames label
G:=SmallGroup(486,188);
// by ID
G=gap.SmallGroup(486,188);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,49,3134,986,4755,303,453,11344,1096,11669]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^9=e^2=1,d^3=c^3,a*b=b*a,a*c=c*a,d*a*d^-1=a*c^3,e*a*e=a*b*c^6,b*c=c*b,b*d=d*b,e*b*e=b^-1,d*c*d^-1=a^-1*b*c,e*c*e=c^-1,e*d*e=c^6*d^2>;
// generators/relations
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