metabelian, supersoluble, monomial
Aliases: (C32×C9)⋊13C6, He3.C3⋊3S3, C3⋊(He3.S3), C32⋊4D9⋊5C3, He3.1(C3⋊S3), (C3×He3).16S3, C33.65(C3×S3), C3.7(He3⋊4S3), C32.18(C32⋊C6), (C3×C9)⋊18(C3×S3), (C3×He3.C3)⋊4C2, C32.17(C3×C3⋊S3), SmallGroup(486,170)
Series: Derived ►Chief ►Lower central ►Upper central
| C32×C9 — C32⋊4D9⋊C3 |
Generators and relations for C32⋊4D9⋊C3
G = < a,b,c,d,e,f | a3=b3=c3=d3=f2=1, e3=fbf=b-1, ab=ba, cac-1=ab-1, ad=da, ae=ea, faf=a-1, bc=cb, bd=db, be=eb, cd=dc, ece-1=a-1bc, cf=fc, de=ed, fdf=d-1, fef=be2 >
Subgroups: 1088 in 96 conjugacy classes, 22 normal (11 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C32, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, C3×C9, He3, He3, 3- 1+2, C33, C33, C32⋊C6, C9⋊S3, C3×C3⋊S3, C33⋊C2, He3.C3, He3.C3, C32×C9, C3×He3, C3×3- 1+2, He3.S3, He3⋊4S3, C32⋊4D9, C3×He3.C3, C32⋊4D9⋊C3
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, C32⋊C6, C3×C3⋊S3, He3.S3, He3⋊4S3, C32⋊4D9⋊C3
Character table of C32⋊4D9⋊C3
| 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 | 2 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | orthogonal lifted from S3 |
| ρ9 | 2 | 0 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ10 | 2 | 0 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 0 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | ζ6 | -1-√-3 | ζ6 | ζ65 | ζ65 | -1+√-3 | complex lifted from C3×S3 |
| ρ12 | 2 | 0 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | ζ6 | ζ6 | -1-√-3 | -1+√-3 | ζ65 | ζ65 | complex lifted from C3×S3 |
| ρ13 | 2 | 0 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | ζ65 | ζ65 | -1+√-3 | -1-√-3 | ζ6 | ζ6 | complex lifted from C3×S3 |
| ρ14 | 2 | 0 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | 0 | 0 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1-√-3 | ζ6 | ζ6 | ζ65 | -1+√-3 | ζ65 | complex lifted from C3×S3 |
| ρ15 | 2 | 0 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | 0 | 0 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1+√-3 | ζ65 | ζ65 | ζ6 | -1-√-3 | ζ6 | complex lifted from C3×S3 |
| ρ16 | 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 |
| ρ17 | 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 |
| ρ18 | 2 | 0 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | ζ65 | -1+√-3 | ζ65 | ζ6 | ζ6 | -1-√-3 | complex lifted from C3×S3 |
| 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 C32⋊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 | 6 | -3 | -3 | -3 | -3 | -3 | 6 | 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 |
| ρ22 | 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 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | 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.S3 |
| ρ23 | 6 | 0 | -3 | -3 | -3 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | 2ζ95+ζ94+ζ92-ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from He3.S3 |
| ρ24 | 6 | 0 | -3 | -3 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ94-ζ92+2ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from He3.S3 |
| ρ25 | 6 | 0 | -3 | -3 | -3 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ94-ζ92+2ζ9 | 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+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | 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.S3 |
| ρ27 | 6 | 0 | -3 | -3 | -3 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ97-ζ94+2ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from He3.S3 |
| ρ28 | 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 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | 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.S3 |
| ρ29 | 6 | 0 | -3 | -3 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | 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 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | 2ζ95+ζ94+ζ92-ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from He3.S3 |
►On 81 points
Generators in S81
(1 79 50)(2 80 51)(3 81 52)(4 73 53)(5 74 54)(6 75 46)(7 76 47)(8 77 48)(9 78 49)(10 63 25)(11 55 26)(12 56 27)(13 57 19)(14 58 20)(15 59 21)(16 60 22)(17 61 23)(18 62 24)(28 69 39)(29 70 40)(30 71 41)(31 72 42)(32 64 43)(33 65 44)(34 66 45)(35 67 37)(36 68 38)
(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)(55 61 58)(56 62 59)(57 63 60)(64 70 67)(65 71 68)(66 72 69)(73 79 76)(74 80 77)(75 81 78)
(2 51 74)(3 81 49)(5 54 77)(6 75 52)(8 48 80)(9 78 46)(10 25 57)(11 55 23)(13 19 60)(14 58 26)(16 22 63)(17 61 20)(21 24 27)(28 45 66)(29 67 40)(30 36 33)(31 39 69)(32 70 43)(34 42 72)(35 64 37)(47 50 53)(56 62 59)(65 68 71)(73 79 76)
(1 38 18)(2 39 10)(3 40 11)(4 41 12)(5 42 13)(6 43 14)(7 44 15)(8 45 16)(9 37 17)(19 54 72)(20 46 64)(21 47 65)(22 48 66)(23 49 67)(24 50 68)(25 51 69)(26 52 70)(27 53 71)(28 63 80)(29 55 81)(30 56 73)(31 57 74)(32 58 75)(33 59 76)(34 60 77)(35 61 78)(36 62 79)
(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 9)(3 8)(4 7)(5 6)(10 37)(11 45)(12 44)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 32)(20 31)(21 30)(22 29)(23 28)(24 36)(25 35)(26 34)(27 33)(46 74)(47 73)(48 81)(49 80)(50 79)(51 78)(52 77)(53 76)(54 75)(55 66)(56 65)(57 64)(58 72)(59 71)(60 70)(61 69)(62 68)(63 67)
G:=sub<Sym(81)| (1,79,50)(2,80,51)(3,81,52)(4,73,53)(5,74,54)(6,75,46)(7,76,47)(8,77,48)(9,78,49)(10,63,25)(11,55,26)(12,56,27)(13,57,19)(14,58,20)(15,59,21)(16,60,22)(17,61,23)(18,62,24)(28,69,39)(29,70,40)(30,71,41)(31,72,42)(32,64,43)(33,65,44)(34,66,45)(35,67,37)(36,68,38), (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)(55,61,58)(56,62,59)(57,63,60)(64,70,67)(65,71,68)(66,72,69)(73,79,76)(74,80,77)(75,81,78), (2,51,74)(3,81,49)(5,54,77)(6,75,52)(8,48,80)(9,78,46)(10,25,57)(11,55,23)(13,19,60)(14,58,26)(16,22,63)(17,61,20)(21,24,27)(28,45,66)(29,67,40)(30,36,33)(31,39,69)(32,70,43)(34,42,72)(35,64,37)(47,50,53)(56,62,59)(65,68,71)(73,79,76), (1,38,18)(2,39,10)(3,40,11)(4,41,12)(5,42,13)(6,43,14)(7,44,15)(8,45,16)(9,37,17)(19,54,72)(20,46,64)(21,47,65)(22,48,66)(23,49,67)(24,50,68)(25,51,69)(26,52,70)(27,53,71)(28,63,80)(29,55,81)(30,56,73)(31,57,74)(32,58,75)(33,59,76)(34,60,77)(35,61,78)(36,62,79), (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), (2,9)(3,8)(4,7)(5,6)(10,37)(11,45)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,32)(20,31)(21,30)(22,29)(23,28)(24,36)(25,35)(26,34)(27,33)(46,74)(47,73)(48,81)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,66)(56,65)(57,64)(58,72)(59,71)(60,70)(61,69)(62,68)(63,67)>;
G:=Group( (1,79,50)(2,80,51)(3,81,52)(4,73,53)(5,74,54)(6,75,46)(7,76,47)(8,77,48)(9,78,49)(10,63,25)(11,55,26)(12,56,27)(13,57,19)(14,58,20)(15,59,21)(16,60,22)(17,61,23)(18,62,24)(28,69,39)(29,70,40)(30,71,41)(31,72,42)(32,64,43)(33,65,44)(34,66,45)(35,67,37)(36,68,38), (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)(55,61,58)(56,62,59)(57,63,60)(64,70,67)(65,71,68)(66,72,69)(73,79,76)(74,80,77)(75,81,78), (2,51,74)(3,81,49)(5,54,77)(6,75,52)(8,48,80)(9,78,46)(10,25,57)(11,55,23)(13,19,60)(14,58,26)(16,22,63)(17,61,20)(21,24,27)(28,45,66)(29,67,40)(30,36,33)(31,39,69)(32,70,43)(34,42,72)(35,64,37)(47,50,53)(56,62,59)(65,68,71)(73,79,76), (1,38,18)(2,39,10)(3,40,11)(4,41,12)(5,42,13)(6,43,14)(7,44,15)(8,45,16)(9,37,17)(19,54,72)(20,46,64)(21,47,65)(22,48,66)(23,49,67)(24,50,68)(25,51,69)(26,52,70)(27,53,71)(28,63,80)(29,55,81)(30,56,73)(31,57,74)(32,58,75)(33,59,76)(34,60,77)(35,61,78)(36,62,79), (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), (2,9)(3,8)(4,7)(5,6)(10,37)(11,45)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,32)(20,31)(21,30)(22,29)(23,28)(24,36)(25,35)(26,34)(27,33)(46,74)(47,73)(48,81)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,66)(56,65)(57,64)(58,72)(59,71)(60,70)(61,69)(62,68)(63,67) );
G=PermutationGroup([[(1,79,50),(2,80,51),(3,81,52),(4,73,53),(5,74,54),(6,75,46),(7,76,47),(8,77,48),(9,78,49),(10,63,25),(11,55,26),(12,56,27),(13,57,19),(14,58,20),(15,59,21),(16,60,22),(17,61,23),(18,62,24),(28,69,39),(29,70,40),(30,71,41),(31,72,42),(32,64,43),(33,65,44),(34,66,45),(35,67,37),(36,68,38)], [(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),(55,61,58),(56,62,59),(57,63,60),(64,70,67),(65,71,68),(66,72,69),(73,79,76),(74,80,77),(75,81,78)], [(2,51,74),(3,81,49),(5,54,77),(6,75,52),(8,48,80),(9,78,46),(10,25,57),(11,55,23),(13,19,60),(14,58,26),(16,22,63),(17,61,20),(21,24,27),(28,45,66),(29,67,40),(30,36,33),(31,39,69),(32,70,43),(34,42,72),(35,64,37),(47,50,53),(56,62,59),(65,68,71),(73,79,76)], [(1,38,18),(2,39,10),(3,40,11),(4,41,12),(5,42,13),(6,43,14),(7,44,15),(8,45,16),(9,37,17),(19,54,72),(20,46,64),(21,47,65),(22,48,66),(23,49,67),(24,50,68),(25,51,69),(26,52,70),(27,53,71),(28,63,80),(29,55,81),(30,56,73),(31,57,74),(32,58,75),(33,59,76),(34,60,77),(35,61,78),(36,62,79)], [(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)], [(2,9),(3,8),(4,7),(5,6),(10,37),(11,45),(12,44),(13,43),(14,42),(15,41),(16,40),(17,39),(18,38),(19,32),(20,31),(21,30),(22,29),(23,28),(24,36),(25,35),(26,34),(27,33),(46,74),(47,73),(48,81),(49,80),(50,79),(51,78),(52,77),(53,76),(54,75),(55,66),(56,65),(57,64),(58,72),(59,71),(60,70),(61,69),(62,68),(63,67)]])
Matrix representation of C32⋊4D9⋊C3 ►in GL8(𝔽19)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 18 | 18 |
| 7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 18 | 18 |
| 0 | 0 | 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 |
| 17 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 18 | 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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 15 | 18 | 15 | 4 | 3 |
| 0 | 0 | 4 | 3 | 4 | 3 | 16 | 1 |
| 0 | 0 | 4 | 3 | 18 | 15 | 18 | 15 |
| 0 | 0 | 16 | 1 | 4 | 3 | 4 | 3 |
| 0 | 0 | 18 | 15 | 4 | 3 | 18 | 15 |
| 0 | 0 | 4 | 3 | 16 | 1 | 4 | 3 |
| 2 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 15 | 18 | 3 | 16 | 15 |
| 0 | 0 | 18 | 3 | 4 | 1 | 18 | 3 |
| 0 | 0 | 18 | 3 | 16 | 15 | 16 | 15 |
| 0 | 0 | 4 | 1 | 18 | 3 | 18 | 3 |
| 0 | 0 | 16 | 15 | 16 | 15 | 18 | 3 |
| 0 | 0 | 18 | 3 | 18 | 3 | 4 | 1 |
G:=sub<GL(8,GF(19))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18],[7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,18,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0],[17,18,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,0,16,17,0,0,0,0,0,0,0,0,18,4,4,16,18,4,0,0,15,3,3,1,15,3,0,0,18,4,18,4,4,16,0,0,15,3,15,3,3,1,0,0,4,16,18,4,18,4,0,0,3,1,15,3,15,3],[2,1,0,0,0,0,0,0,16,17,0,0,0,0,0,0,0,0,16,18,18,4,16,18,0,0,15,3,3,1,15,3,0,0,18,4,16,18,16,18,0,0,3,1,15,3,15,3,0,0,16,18,16,18,18,4,0,0,15,3,15,3,3,1] >;
C32⋊4D9⋊C3 in GAP, Magma, Sage, TeX
C_3^2\rtimes_4D_9\rtimes C_3
% in TeX
G:=Group("C3^2:4D9:C3"); // GroupNames label
G:=SmallGroup(486,170);
// by ID
G=gap.SmallGroup(486,170);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,6050,548,500,867,3244,3250,11669]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^3=f^2=1,e^3=f*b*f=b^-1,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,a*e=e*a,f*a*f=a^-1,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=a^-1*b*c,c*f=f*c,d*e=e*d,f*d*f=d^-1,f*e*f=b*e^2>;
// generators/relations
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