metabelian, supersoluble, monomial
Aliases: He3⋊1D9, He3⋊C9⋊2C2, (C32×C9)⋊5C6, (C3×He3).3S3, C32.1(C3×D9), C32⋊4D9⋊1C3, C3.(C33⋊C6), C32.1(C9⋊C6), C33.52(C3×S3), C3.(He3.S3), C3.2(C32⋊D9), C32.40(C32⋊C6), C3.1(He3.2S3), SmallGroup(486,25)
Series: Derived ►Chief ►Lower central ►Upper central
| C32×C9 — He3⋊D9 |
Generators and relations for He3⋊D9
G = < a,b,c,d,e | a3=b3=c3=d9=e2=1, ab=ba, cac-1=ab-1, ad=da, eae=a-1, bc=cb, bd=db, ebe=b-1, dcd-1=ece=ab-1c, ede=d-1 >
Subgroups: 1052 in 76 conjugacy classes, 16 normal (14 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, He3, He3, C33, C33, C32⋊C6, C9⋊S3, C3×C3⋊S3, C33⋊C2, C32⋊C9, C32×C9, C3×He3, He3⋊4S3, C32⋊4D9, He3⋊C9, He3⋊D9
Quotients: C1, C2, C3, S3, C6, D9, C3×S3, C3×D9, C32⋊C6, C9⋊C6, C32⋊D9, C33⋊C6, He3.S3, He3.2S3, He3⋊D9
Character table of He3⋊D9
| 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 | 6 | 6 | 6 | 9 | 9 | 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 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | linear of order 6 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | linear of order 3 |
| ρ6 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | linear of order 6 |
| ρ7 | 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 |
| ρ8 | 2 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ9 | 2 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ10 | 2 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ11 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1-√-3 | -1+√-3 | -1-√-3 | -1+√-3 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | complex lifted from C3×S3 |
| ρ12 | 2 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ94 | ζ92+ζ9 | ζ97+ζ95 | ζ94+ζ92 | ζ95+ζ9 | ζ98+ζ97 | complex lifted from C3×D9 |
| ρ13 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1+√-3 | -1-√-3 | -1+√-3 | -1-√-3 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | complex lifted from C3×S3 |
| ρ14 | 2 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ94+ζ92 | ζ95+ζ9 | ζ98+ζ97 | ζ92+ζ9 | ζ97+ζ95 | ζ98+ζ94 | complex lifted from C3×D9 |
| ρ15 | 2 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ92+ζ9 | ζ97+ζ95 | ζ98+ζ94 | ζ95+ζ9 | ζ98+ζ97 | ζ94+ζ92 | complex lifted from C3×D9 |
| ρ16 | 2 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ97 | ζ94+ζ92 | ζ95+ζ9 | ζ98+ζ94 | ζ92+ζ9 | ζ97+ζ95 | complex lifted from C3×D9 |
| ρ17 | 2 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ9 | ζ98+ζ97 | ζ94+ζ92 | ζ97+ζ95 | ζ98+ζ94 | ζ92+ζ9 | complex lifted from C3×D9 |
| ρ18 | 2 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ95 | ζ98+ζ94 | ζ92+ζ9 | ζ98+ζ97 | ζ94+ζ92 | ζ95+ζ9 | complex lifted from C3×D9 |
| ρ19 | 6 | 0 | 6 | -3 | -3 | -3 | 6 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C9⋊C6 |
| ρ20 | 6 | 0 | 6 | 6 | 6 | 6 | -3 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C6 |
| ρ21 | 6 | 0 | -3 | -3 | -3 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 | -3 | -3 | -3 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C33⋊C6 |
| ρ22 | 6 | 0 | -3 | -3 | -3 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 | 3 | -3 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C33⋊C6 |
| ρ23 | 6 | 0 | 6 | -3 | -3 | -3 | -3 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C9⋊C6 |
| ρ24 | 6 | 0 | -3 | -3 | -3 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3 | -3 | 0 | 0 | 0 | 3 | 3 | 3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C33⋊C6 |
| ρ25 | 6 | 0 | -3 | 6 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ97-ζ94+2ζ92 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ97-ζ94+2ζ92 | 2ζ95+ζ94+ζ92-ζ9 | 2ζ95+ζ94+ζ92-ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from He3.S3 |
| ρ26 | 6 | 0 | -3 | 6 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ98+ζ97-ζ94+2ζ92 | 2ζ95+ζ94+ζ92-ζ9 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ97-ζ94+2ζ92 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ94-ζ92+2ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from He3.S3 |
| ρ27 | 6 | 0 | -3 | -3 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from He3.2S3 |
| ρ28 | 6 | 0 | -3 | -3 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from He3.2S3 |
| ρ29 | 6 | 0 | -3 | 6 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ97-ζ94+2ζ92 | 2ζ95+ζ94+ζ92-ζ9 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ97-ζ94+2ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from He3.S3 |
| ρ30 | 6 | 0 | -3 | -3 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from He3.2S3 |
►On 81 points
Generators in S81
(1 4 7)(2 5 8)(3 6 9)(10 79 20)(11 80 21)(12 81 22)(13 73 23)(14 74 24)(15 75 25)(16 76 26)(17 77 27)(18 78 19)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 64 62)(47 65 63)(48 66 55)(49 67 56)(50 68 57)(51 69 58)(52 70 59)(53 71 60)(54 72 61)
(1 39 36)(2 40 28)(3 41 29)(4 42 30)(5 43 31)(6 44 32)(7 45 33)(8 37 34)(9 38 35)(10 23 76)(11 24 77)(12 25 78)(13 26 79)(14 27 80)(15 19 81)(16 20 73)(17 21 74)(18 22 75)(46 70 56)(47 71 57)(48 72 58)(49 64 59)(50 65 60)(51 66 61)(52 67 62)(53 68 63)(54 69 55)
(1 22 52)(2 13 50)(3 74 48)(4 25 46)(5 16 53)(6 77 51)(7 19 49)(8 10 47)(9 80 54)(11 66 44)(12 56 30)(14 69 38)(15 59 33)(17 72 41)(18 62 36)(20 68 43)(21 58 29)(23 71 37)(24 61 32)(26 65 40)(27 55 35)(28 79 60)(31 73 63)(34 76 57)(39 75 67)(42 78 70)(45 81 64)
(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 9)(2 8)(3 7)(4 6)(10 73)(11 81)(12 80)(13 79)(14 78)(15 77)(16 76)(17 75)(18 74)(19 24)(20 23)(21 22)(25 27)(28 37)(29 45)(30 44)(31 43)(32 42)(33 41)(34 40)(35 39)(36 38)(46 48)(49 54)(50 53)(51 52)(55 64)(56 72)(57 71)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)
G:=sub<Sym(81)| (1,4,7)(2,5,8)(3,6,9)(10,79,20)(11,80,21)(12,81,22)(13,73,23)(14,74,24)(15,75,25)(16,76,26)(17,77,27)(18,78,19)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,64,62)(47,65,63)(48,66,55)(49,67,56)(50,68,57)(51,69,58)(52,70,59)(53,71,60)(54,72,61), (1,39,36)(2,40,28)(3,41,29)(4,42,30)(5,43,31)(6,44,32)(7,45,33)(8,37,34)(9,38,35)(10,23,76)(11,24,77)(12,25,78)(13,26,79)(14,27,80)(15,19,81)(16,20,73)(17,21,74)(18,22,75)(46,70,56)(47,71,57)(48,72,58)(49,64,59)(50,65,60)(51,66,61)(52,67,62)(53,68,63)(54,69,55), (1,22,52)(2,13,50)(3,74,48)(4,25,46)(5,16,53)(6,77,51)(7,19,49)(8,10,47)(9,80,54)(11,66,44)(12,56,30)(14,69,38)(15,59,33)(17,72,41)(18,62,36)(20,68,43)(21,58,29)(23,71,37)(24,61,32)(26,65,40)(27,55,35)(28,79,60)(31,73,63)(34,76,57)(39,75,67)(42,78,70)(45,81,64), (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), (1,9)(2,8)(3,7)(4,6)(10,73)(11,81)(12,80)(13,79)(14,78)(15,77)(16,76)(17,75)(18,74)(19,24)(20,23)(21,22)(25,27)(28,37)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(46,48)(49,54)(50,53)(51,52)(55,64)(56,72)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)>;
G:=Group( (1,4,7)(2,5,8)(3,6,9)(10,79,20)(11,80,21)(12,81,22)(13,73,23)(14,74,24)(15,75,25)(16,76,26)(17,77,27)(18,78,19)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,64,62)(47,65,63)(48,66,55)(49,67,56)(50,68,57)(51,69,58)(52,70,59)(53,71,60)(54,72,61), (1,39,36)(2,40,28)(3,41,29)(4,42,30)(5,43,31)(6,44,32)(7,45,33)(8,37,34)(9,38,35)(10,23,76)(11,24,77)(12,25,78)(13,26,79)(14,27,80)(15,19,81)(16,20,73)(17,21,74)(18,22,75)(46,70,56)(47,71,57)(48,72,58)(49,64,59)(50,65,60)(51,66,61)(52,67,62)(53,68,63)(54,69,55), (1,22,52)(2,13,50)(3,74,48)(4,25,46)(5,16,53)(6,77,51)(7,19,49)(8,10,47)(9,80,54)(11,66,44)(12,56,30)(14,69,38)(15,59,33)(17,72,41)(18,62,36)(20,68,43)(21,58,29)(23,71,37)(24,61,32)(26,65,40)(27,55,35)(28,79,60)(31,73,63)(34,76,57)(39,75,67)(42,78,70)(45,81,64), (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), (1,9)(2,8)(3,7)(4,6)(10,73)(11,81)(12,80)(13,79)(14,78)(15,77)(16,76)(17,75)(18,74)(19,24)(20,23)(21,22)(25,27)(28,37)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(46,48)(49,54)(50,53)(51,52)(55,64)(56,72)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65) );
G=PermutationGroup([[(1,4,7),(2,5,8),(3,6,9),(10,79,20),(11,80,21),(12,81,22),(13,73,23),(14,74,24),(15,75,25),(16,76,26),(17,77,27),(18,78,19),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,64,62),(47,65,63),(48,66,55),(49,67,56),(50,68,57),(51,69,58),(52,70,59),(53,71,60),(54,72,61)], [(1,39,36),(2,40,28),(3,41,29),(4,42,30),(5,43,31),(6,44,32),(7,45,33),(8,37,34),(9,38,35),(10,23,76),(11,24,77),(12,25,78),(13,26,79),(14,27,80),(15,19,81),(16,20,73),(17,21,74),(18,22,75),(46,70,56),(47,71,57),(48,72,58),(49,64,59),(50,65,60),(51,66,61),(52,67,62),(53,68,63),(54,69,55)], [(1,22,52),(2,13,50),(3,74,48),(4,25,46),(5,16,53),(6,77,51),(7,19,49),(8,10,47),(9,80,54),(11,66,44),(12,56,30),(14,69,38),(15,59,33),(17,72,41),(18,62,36),(20,68,43),(21,58,29),(23,71,37),(24,61,32),(26,65,40),(27,55,35),(28,79,60),(31,73,63),(34,76,57),(39,75,67),(42,78,70),(45,81,64)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,73),(11,81),(12,80),(13,79),(14,78),(15,77),(16,76),(17,75),(18,74),(19,24),(20,23),(21,22),(25,27),(28,37),(29,45),(30,44),(31,43),(32,42),(33,41),(34,40),(35,39),(36,38),(46,48),(49,54),(50,53),(51,52),(55,64),(56,72),(57,71),(58,70),(59,69),(60,68),(61,67),(62,66),(63,65)]])
Matrix representation of He3⋊D9 ►in GL8(𝔽19)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 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 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 18 | 18 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 7 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 7 | 18 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 14 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 | 7 |
| 7 | 18 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 14 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 17 |
G:=sub<GL(8,GF(19))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,18,0],[7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[7,1,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,2,5,0,0,0,0,0,0,14,7,0,0,0,0,0,0,0,0,12,2,0,0,0,0,0,0,17,14,0,0,0,0,0,0,0,0,2,5,0,0,0,0,0,0,14,7],[7,10,0,0,0,0,0,0,18,12,0,0,0,0,0,0,0,0,2,12,0,0,0,0,0,0,14,17,0,0,0,0,0,0,0,0,12,5,0,0,0,0,0,0,17,7,0,0,0,0,0,0,0,0,2,12,0,0,0,0,0,0,14,17] >;
He3⋊D9 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes D_9 % in TeX
G:=Group("He3:D9"); // GroupNames label
G:=SmallGroup(486,25);
// by ID
G=gap.SmallGroup(486,25);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,3134,224,986,867,873,3244,11669]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^9=e^2=1,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,e*a*e=a^-1,b*c=c*b,b*d=d*b,e*b*e=b^-1,d*c*d^-1=e*c*e=a*b^-1*c,e*d*e=d^-1>;
// generators/relations
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