non-abelian, supersoluble, monomial
Aliases: (C32×C9)⋊19S3, He3.C3⋊1S3, He3.4(C3⋊S3), C3⋊(He3.3S3), (C3×He3).19S3, C33.38(C3⋊S3), C3.4(He3⋊5S3), 3- 1+2⋊2(C3⋊S3), (C3×3- 1+2)⋊11S3, C32.2(C33⋊C2), C32.28(He3⋊C2), (C3×C9)⋊5(C3⋊S3), (C3×He3.C3)⋊2C2, SmallGroup(486,186)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×He3.C3 — He3.(C3⋊S3) |
Generators and relations for He3.(C3⋊S3)
G = < a,b,c,d,e | a3=b3=c9=d3=e2=1, ab=ba, ac=ca, dad-1=ac6, eae=abc3, bc=cb, bd=db, ebe=b-1, dcd-1=a-1bc, ece=c-1, ede=d-1 >
Subgroups: 1168 in 126 conjugacy classes, 35 normal (9 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, C3×C9, He3, He3, 3- 1+2, 3- 1+2, C33, C33, C3×D9, C32⋊C6, C9⋊C6, C9⋊S3, C3×C3⋊S3, C33⋊C2, He3.C3, C32×C9, C3×He3, C3×3- 1+2, He3.3S3, C3×C9⋊S3, He3⋊4S3, C33.S3, C3×He3.C3, He3.(C3⋊S3)
Quotients: C1, C2, S3, C3⋊S3, He3⋊C2, C33⋊C2, He3.3S3, He3⋊5S3, He3.(C3⋊S3)
Character table of He3.(C3⋊S3)
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 3I | 3J | 3K | 6A | 6B | 9A | 9B | 9C | 9D | 9E | 9F | 9G | 9H | 9I | 9J | 9K | 9L | 9M | 9N | 9O | |
| size | 1 | 81 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 18 | 18 | 18 | 81 | 81 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 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 | -1 | -1 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ4 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ5 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 2 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | -1 | 2 | orthogonal lifted from S3 |
| ρ7 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | 2 | -1 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 2 | -1 | -1 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ9 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | 2 | -1 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ11 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 2 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | 2 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | 2 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | 2 | 2 | orthogonal lifted from S3 |
| ρ15 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | 2 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | 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 | 0 | 0 | 0 | ζ65 | ζ6 | 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 | 0 | 0 | 0 | ζ32 | ζ3 | 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 | 0 | 0 | 0 | ζ3 | ζ32 | 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 | 0 | 0 | 0 | ζ6 | ζ65 | 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 | 0 | 0 | 0 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from He3.3S3 |
| ρ21 | 6 | 0 | -3 | -3 | -3 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ97-ζ94+2ζ92 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from He3.3S3 |
| ρ22 | 6 | 0 | -3 | 6 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ97-ζ94+2ζ92 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from He3.3S3 |
| ρ23 | 6 | 0 | -3 | -3 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ97-ζ94+2ζ92 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from He3.3S3 |
| ρ24 | 6 | 0 | -3 | 6 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ97-ζ94+2ζ92 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from He3.3S3 |
| ρ25 | 6 | 0 | -3 | -3 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from He3.3S3 |
| ρ26 | 6 | 0 | -3 | -3 | -3 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from He3.3S3 |
| ρ27 | 6 | 0 | -3 | -3 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ97-ζ94+2ζ92 | 2ζ95+ζ94+ζ92-ζ9 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from He3.3S3 |
| ρ28 | 6 | 0 | -3 | -3 | -3 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ97-ζ94+2ζ92 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from He3.3S3 |
| 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 62 23)(2 63 24)(3 55 25)(4 56 26)(5 57 27)(6 58 19)(7 59 20)(8 60 21)(9 61 22)(10 37 47)(11 38 48)(12 39 49)(13 40 50)(14 41 51)(15 42 52)(16 43 53)(17 44 54)(18 45 46)(28 67 77)(29 68 78)(30 69 79)(31 70 80)(32 71 81)(33 72 73)(34 64 74)(35 65 75)(36 66 76)
(1 48 75)(2 49 76)(3 50 77)(4 51 78)(5 52 79)(6 53 80)(7 54 81)(8 46 73)(9 47 74)(10 34 61)(11 35 62)(12 36 63)(13 28 55)(14 29 56)(15 30 57)(16 31 58)(17 32 59)(18 33 60)(19 43 70)(20 44 71)(21 45 72)(22 37 64)(23 38 65)(24 39 66)(25 40 67)(26 41 68)(27 42 69)
(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)
(2 30 39)(3 37 28)(5 33 42)(6 40 31)(8 36 45)(9 43 34)(10 74 19)(11 14 17)(12 21 73)(13 77 22)(15 24 76)(16 80 25)(18 27 79)(20 26 23)(29 32 35)(38 44 41)(46 63 72)(47 70 61)(49 57 66)(50 64 55)(52 60 69)(53 67 58)(56 59 62)(65 71 68)
(2 9)(3 8)(4 7)(5 6)(10 57)(11 56)(12 55)(13 63)(14 62)(15 61)(16 60)(17 59)(18 58)(19 66)(20 65)(21 64)(22 72)(23 71)(24 70)(25 69)(26 68)(27 67)(28 36)(29 35)(30 34)(31 33)(37 45)(38 44)(39 43)(40 42)(46 77)(47 76)(48 75)(49 74)(50 73)(51 81)(52 80)(53 79)(54 78)
G:=sub<Sym(81)| (1,62,23)(2,63,24)(3,55,25)(4,56,26)(5,57,27)(6,58,19)(7,59,20)(8,60,21)(9,61,22)(10,37,47)(11,38,48)(12,39,49)(13,40,50)(14,41,51)(15,42,52)(16,43,53)(17,44,54)(18,45,46)(28,67,77)(29,68,78)(30,69,79)(31,70,80)(32,71,81)(33,72,73)(34,64,74)(35,65,75)(36,66,76), (1,48,75)(2,49,76)(3,50,77)(4,51,78)(5,52,79)(6,53,80)(7,54,81)(8,46,73)(9,47,74)(10,34,61)(11,35,62)(12,36,63)(13,28,55)(14,29,56)(15,30,57)(16,31,58)(17,32,59)(18,33,60)(19,43,70)(20,44,71)(21,45,72)(22,37,64)(23,38,65)(24,39,66)(25,40,67)(26,41,68)(27,42,69), (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), (2,30,39)(3,37,28)(5,33,42)(6,40,31)(8,36,45)(9,43,34)(10,74,19)(11,14,17)(12,21,73)(13,77,22)(15,24,76)(16,80,25)(18,27,79)(20,26,23)(29,32,35)(38,44,41)(46,63,72)(47,70,61)(49,57,66)(50,64,55)(52,60,69)(53,67,58)(56,59,62)(65,71,68), (2,9)(3,8)(4,7)(5,6)(10,57)(11,56)(12,55)(13,63)(14,62)(15,61)(16,60)(17,59)(18,58)(19,66)(20,65)(21,64)(22,72)(23,71)(24,70)(25,69)(26,68)(27,67)(28,36)(29,35)(30,34)(31,33)(37,45)(38,44)(39,43)(40,42)(46,77)(47,76)(48,75)(49,74)(50,73)(51,81)(52,80)(53,79)(54,78)>;
G:=Group( (1,62,23)(2,63,24)(3,55,25)(4,56,26)(5,57,27)(6,58,19)(7,59,20)(8,60,21)(9,61,22)(10,37,47)(11,38,48)(12,39,49)(13,40,50)(14,41,51)(15,42,52)(16,43,53)(17,44,54)(18,45,46)(28,67,77)(29,68,78)(30,69,79)(31,70,80)(32,71,81)(33,72,73)(34,64,74)(35,65,75)(36,66,76), (1,48,75)(2,49,76)(3,50,77)(4,51,78)(5,52,79)(6,53,80)(7,54,81)(8,46,73)(9,47,74)(10,34,61)(11,35,62)(12,36,63)(13,28,55)(14,29,56)(15,30,57)(16,31,58)(17,32,59)(18,33,60)(19,43,70)(20,44,71)(21,45,72)(22,37,64)(23,38,65)(24,39,66)(25,40,67)(26,41,68)(27,42,69), (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), (2,30,39)(3,37,28)(5,33,42)(6,40,31)(8,36,45)(9,43,34)(10,74,19)(11,14,17)(12,21,73)(13,77,22)(15,24,76)(16,80,25)(18,27,79)(20,26,23)(29,32,35)(38,44,41)(46,63,72)(47,70,61)(49,57,66)(50,64,55)(52,60,69)(53,67,58)(56,59,62)(65,71,68), (2,9)(3,8)(4,7)(5,6)(10,57)(11,56)(12,55)(13,63)(14,62)(15,61)(16,60)(17,59)(18,58)(19,66)(20,65)(21,64)(22,72)(23,71)(24,70)(25,69)(26,68)(27,67)(28,36)(29,35)(30,34)(31,33)(37,45)(38,44)(39,43)(40,42)(46,77)(47,76)(48,75)(49,74)(50,73)(51,81)(52,80)(53,79)(54,78) );
G=PermutationGroup([[(1,62,23),(2,63,24),(3,55,25),(4,56,26),(5,57,27),(6,58,19),(7,59,20),(8,60,21),(9,61,22),(10,37,47),(11,38,48),(12,39,49),(13,40,50),(14,41,51),(15,42,52),(16,43,53),(17,44,54),(18,45,46),(28,67,77),(29,68,78),(30,69,79),(31,70,80),(32,71,81),(33,72,73),(34,64,74),(35,65,75),(36,66,76)], [(1,48,75),(2,49,76),(3,50,77),(4,51,78),(5,52,79),(6,53,80),(7,54,81),(8,46,73),(9,47,74),(10,34,61),(11,35,62),(12,36,63),(13,28,55),(14,29,56),(15,30,57),(16,31,58),(17,32,59),(18,33,60),(19,43,70),(20,44,71),(21,45,72),(22,37,64),(23,38,65),(24,39,66),(25,40,67),(26,41,68),(27,42,69)], [(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)], [(2,30,39),(3,37,28),(5,33,42),(6,40,31),(8,36,45),(9,43,34),(10,74,19),(11,14,17),(12,21,73),(13,77,22),(15,24,76),(16,80,25),(18,27,79),(20,26,23),(29,32,35),(38,44,41),(46,63,72),(47,70,61),(49,57,66),(50,64,55),(52,60,69),(53,67,58),(56,59,62),(65,71,68)], [(2,9),(3,8),(4,7),(5,6),(10,57),(11,56),(12,55),(13,63),(14,62),(15,61),(16,60),(17,59),(18,58),(19,66),(20,65),(21,64),(22,72),(23,71),(24,70),(25,69),(26,68),(27,67),(28,36),(29,35),(30,34),(31,33),(37,45),(38,44),(39,43),(40,42),(46,77),(47,76),(48,75),(49,74),(50,73),(51,81),(52,80),(53,79),(54,78)]])
Matrix representation of He3.(C3⋊S3) ►in GL8(𝔽19)
| 0 | 18 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 18 | 0 | 0 | 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 | 18 | 1 |
| 0 | 0 | 12 | 12 | 8 | 8 | 17 | 18 |
| 0 | 0 | 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 11 | 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 | 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 | 1 | 13 | 8 | 1 | 12 | 17 |
| 0 | 0 | 14 | 9 | 17 | 12 | 14 | 12 |
| 0 | 0 | 10 | 12 | 9 | 2 | 5 | 7 |
| 0 | 0 | 4 | 11 | 15 | 10 | 17 | 5 |
| 0 | 0 | 1 | 0 | 7 | 0 | 16 | 11 |
| 0 | 0 | 13 | 5 | 14 | 9 | 2 | 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 | 18 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 8 | 18 | 18 |
| 0 | 0 | 0 | 0 | 11 | 0 | 1 | 0 |
| 18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 18 | 1 | 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 | 1 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 11 | 1 | 0 |
G:=sub<GL(8,GF(19))| [0,1,0,0,0,0,0,0,18,18,0,0,0,0,0,0,0,0,0,0,0,12,0,1,0,0,0,0,0,12,0,0,0,0,1,0,0,8,0,0,0,0,0,1,0,8,0,0,0,0,0,0,18,17,11,11,0,0,0,0,1,18,0,0],[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],[18,18,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,14,10,4,1,13,0,0,13,9,12,11,0,5,0,0,8,17,9,15,7,14,0,0,1,12,2,10,0,9,0,0,12,14,5,17,16,2,0,0,17,12,7,5,11,18],[0,1,0,0,0,0,0,0,18,18,0,0,0,0,0,0,0,0,1,0,0,0,12,0,0,0,0,1,0,0,12,0,0,0,0,0,18,18,0,11,0,0,0,0,1,0,8,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,18,0],[18,18,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,18,18,11,11,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
He3.(C3⋊S3) in GAP, Magma, Sage, TeX
{\rm He}_3.(C_3\rtimes S_3) % in TeX
G:=Group("He3.(C3:S3)"); // GroupNames label
G:=SmallGroup(486,186);
// by ID
G=gap.SmallGroup(486,186);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,49,218,8643,303,237,11344,1096,11669]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^9=d^3=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*c^6,e*a*e=a*b*c^3,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=d^-1>;
// generators/relations
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