G = C3⋊(He3⋊S3) order 486 = 2·35
The semidirect product of C3 and He3⋊S3 acting via He3⋊S3/He3⋊C3=C2
non-abelian, supersoluble, monomial
Aliases: C3⋊(He3⋊S3), He3⋊3(C3⋊S3), (C3×He3)⋊16S3, (C32×C9)⋊20S3, He3⋊C3⋊5S3, C33.39(C3⋊S3), C3.5(He3⋊5S3), C32.3(C33⋊C2), C32.29(He3⋊C2), (C3×C9)⋊6(C3⋊S3), (C3×He3⋊C3)⋊7C2, SmallGroup(486,187)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×He3⋊C3 — C3⋊(He3⋊S3) |
Generators and relations for C3⋊(He3⋊S3)
G = < a,b,c,d,e,f | a3=b3=c3=d3=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, faf=a-1, ebe-1=bc=cb, dbd-1=bc-1, bf=fb, cd=dc, ce=ec, fcf=c-1, ede-1=b-1c-1d, fdf=bc-1d-1, fef=e-1 >
Subgroups: 1924 in 150 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, He3, He3, C33, C33, C3×D9, C32⋊C6, C9⋊S3, C3×C3⋊S3, C33⋊C2, He3⋊C3, C32×C9, C3×He3, He3⋊S3, C3×C9⋊S3, He3⋊4S3, C3×He3⋊C3, C3⋊(He3⋊S3)
Quotients: C1, C2, S3, C3⋊S3, He3⋊C2, C33⋊C2, He3⋊S3, He3⋊5S3, C3⋊(He3⋊S3)
Character table of C3⋊(He3⋊S3)
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 3I | 3J | 3K | 3L | 3M | 3N | 3O | 3P | 3Q | 6A | 6B | 9A | 9B | 9C | 9D | 9E | 9F | 9G | 9H | 9I | |
| size | 1 | 81 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 81 | 81 | 6 | 6 | 6 | 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 | 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 | 2 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ4 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ5 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 2 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 2 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | 2 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ9 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | 2 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | 2 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 2 | -1 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ13 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | orthogonal lifted from S3 |
| ρ14 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | 2 | -1 | orthogonal lifted from S3 |
| ρ15 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | -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 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3 | ζ32 | 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 | 0 | 0 | 0 | 0 | 0 | 0 | ζ32 | ζ3 | 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 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | 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 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | 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 | 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 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | orthogonal lifted from He3⋊S3 |
| ρ21 | 6 | 0 | -3 | -3 | -3 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | orthogonal lifted from He3⋊S3 |
| ρ22 | 6 | 0 | -3 | -3 | -3 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | orthogonal lifted from He3⋊S3 |
| ρ23 | 6 | 0 | -3 | -3 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ98-ζ94+ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | orthogonal lifted from He3⋊S3 |
| ρ24 | 6 | 0 | -3 | -3 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ95+2ζ94-ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | orthogonal lifted from He3⋊S3 |
| ρ25 | 6 | 0 | -3 | 6 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 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 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | orthogonal lifted from He3⋊S3 |
| ρ26 | 6 | 0 | -3 | -3 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ98+2ζ97+ζ94+ζ92 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | orthogonal lifted from He3⋊S3 |
| ρ27 | 6 | 0 | -3 | -3 | -3 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | -ζ98+2ζ97+ζ94+ζ92 | orthogonal lifted from He3⋊S3 |
| ρ28 | 6 | 0 | -3 | 6 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 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 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | orthogonal lifted from He3⋊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 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 21 58)(2 19 59)(3 20 60)(4 53 39)(5 54 37)(6 52 38)(7 36 47)(8 34 48)(9 35 46)(10 23 69)(11 24 67)(12 22 68)(13 41 77)(14 42 78)(15 40 76)(16 31 75)(17 32 73)(18 33 74)(25 70 62)(26 71 63)(27 72 61)(28 57 79)(29 55 80)(30 56 81)(43 49 65)(44 50 66)(45 51 64)
(1 73 38)(2 74 39)(3 75 37)(4 19 18)(5 20 16)(6 21 17)(7 78 11)(8 76 12)(9 77 10)(13 23 35)(14 24 36)(15 22 34)(25 50 81)(26 51 79)(27 49 80)(28 71 64)(29 72 65)(30 70 66)(31 54 60)(32 52 58)(33 53 59)(40 68 48)(41 69 46)(42 67 47)(43 55 61)(44 56 62)(45 57 63)
(1 14 65)(2 15 66)(3 13 64)(4 40 56)(5 41 57)(6 42 55)(7 80 58)(8 81 59)(9 79 60)(10 51 54)(11 49 52)(12 50 53)(16 46 45)(17 47 43)(18 48 44)(19 68 62)(20 69 63)(21 67 61)(22 30 74)(23 28 75)(24 29 73)(25 33 76)(26 31 77)(27 32 78)(34 70 39)(35 71 37)(36 72 38)
(1 22 45)(2 23 43)(3 24 44)(4 46 49)(5 47 50)(6 48 51)(7 70 60)(8 71 58)(9 72 59)(10 29 53)(11 30 54)(12 28 52)(13 61 39)(14 62 37)(15 63 38)(16 67 25)(17 68 26)(18 69 27)(19 41 80)(20 42 81)(21 40 79)(31 78 66)(32 76 64)(33 77 65)(34 57 73)(35 55 74)(36 56 75)
(2 3)(4 16)(5 18)(6 17)(7 72)(8 71)(9 70)(10 66)(11 65)(12 64)(13 56)(14 55)(15 57)(19 20)(22 45)(23 44)(24 43)(25 46)(26 48)(27 47)(28 76)(29 78)(30 77)(31 53)(32 52)(33 54)(34 63)(35 62)(36 61)(37 74)(38 73)(39 75)(40 79)(41 81)(42 80)(49 67)(50 69)(51 68)(59 60)
G:=sub<Sym(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), (1,21,58)(2,19,59)(3,20,60)(4,53,39)(5,54,37)(6,52,38)(7,36,47)(8,34,48)(9,35,46)(10,23,69)(11,24,67)(12,22,68)(13,41,77)(14,42,78)(15,40,76)(16,31,75)(17,32,73)(18,33,74)(25,70,62)(26,71,63)(27,72,61)(28,57,79)(29,55,80)(30,56,81)(43,49,65)(44,50,66)(45,51,64), (1,73,38)(2,74,39)(3,75,37)(4,19,18)(5,20,16)(6,21,17)(7,78,11)(8,76,12)(9,77,10)(13,23,35)(14,24,36)(15,22,34)(25,50,81)(26,51,79)(27,49,80)(28,71,64)(29,72,65)(30,70,66)(31,54,60)(32,52,58)(33,53,59)(40,68,48)(41,69,46)(42,67,47)(43,55,61)(44,56,62)(45,57,63), (1,14,65)(2,15,66)(3,13,64)(4,40,56)(5,41,57)(6,42,55)(7,80,58)(8,81,59)(9,79,60)(10,51,54)(11,49,52)(12,50,53)(16,46,45)(17,47,43)(18,48,44)(19,68,62)(20,69,63)(21,67,61)(22,30,74)(23,28,75)(24,29,73)(25,33,76)(26,31,77)(27,32,78)(34,70,39)(35,71,37)(36,72,38), (1,22,45)(2,23,43)(3,24,44)(4,46,49)(5,47,50)(6,48,51)(7,70,60)(8,71,58)(9,72,59)(10,29,53)(11,30,54)(12,28,52)(13,61,39)(14,62,37)(15,63,38)(16,67,25)(17,68,26)(18,69,27)(19,41,80)(20,42,81)(21,40,79)(31,78,66)(32,76,64)(33,77,65)(34,57,73)(35,55,74)(36,56,75), (2,3)(4,16)(5,18)(6,17)(7,72)(8,71)(9,70)(10,66)(11,65)(12,64)(13,56)(14,55)(15,57)(19,20)(22,45)(23,44)(24,43)(25,46)(26,48)(27,47)(28,76)(29,78)(30,77)(31,53)(32,52)(33,54)(34,63)(35,62)(36,61)(37,74)(38,73)(39,75)(40,79)(41,81)(42,80)(49,67)(50,69)(51,68)(59,60)>;
G:=Group( (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), (1,21,58)(2,19,59)(3,20,60)(4,53,39)(5,54,37)(6,52,38)(7,36,47)(8,34,48)(9,35,46)(10,23,69)(11,24,67)(12,22,68)(13,41,77)(14,42,78)(15,40,76)(16,31,75)(17,32,73)(18,33,74)(25,70,62)(26,71,63)(27,72,61)(28,57,79)(29,55,80)(30,56,81)(43,49,65)(44,50,66)(45,51,64), (1,73,38)(2,74,39)(3,75,37)(4,19,18)(5,20,16)(6,21,17)(7,78,11)(8,76,12)(9,77,10)(13,23,35)(14,24,36)(15,22,34)(25,50,81)(26,51,79)(27,49,80)(28,71,64)(29,72,65)(30,70,66)(31,54,60)(32,52,58)(33,53,59)(40,68,48)(41,69,46)(42,67,47)(43,55,61)(44,56,62)(45,57,63), (1,14,65)(2,15,66)(3,13,64)(4,40,56)(5,41,57)(6,42,55)(7,80,58)(8,81,59)(9,79,60)(10,51,54)(11,49,52)(12,50,53)(16,46,45)(17,47,43)(18,48,44)(19,68,62)(20,69,63)(21,67,61)(22,30,74)(23,28,75)(24,29,73)(25,33,76)(26,31,77)(27,32,78)(34,70,39)(35,71,37)(36,72,38), (1,22,45)(2,23,43)(3,24,44)(4,46,49)(5,47,50)(6,48,51)(7,70,60)(8,71,58)(9,72,59)(10,29,53)(11,30,54)(12,28,52)(13,61,39)(14,62,37)(15,63,38)(16,67,25)(17,68,26)(18,69,27)(19,41,80)(20,42,81)(21,40,79)(31,78,66)(32,76,64)(33,77,65)(34,57,73)(35,55,74)(36,56,75), (2,3)(4,16)(5,18)(6,17)(7,72)(8,71)(9,70)(10,66)(11,65)(12,64)(13,56)(14,55)(15,57)(19,20)(22,45)(23,44)(24,43)(25,46)(26,48)(27,47)(28,76)(29,78)(30,77)(31,53)(32,52)(33,54)(34,63)(35,62)(36,61)(37,74)(38,73)(39,75)(40,79)(41,81)(42,80)(49,67)(50,69)(51,68)(59,60) );
G=PermutationGroup([[(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)], [(1,21,58),(2,19,59),(3,20,60),(4,53,39),(5,54,37),(6,52,38),(7,36,47),(8,34,48),(9,35,46),(10,23,69),(11,24,67),(12,22,68),(13,41,77),(14,42,78),(15,40,76),(16,31,75),(17,32,73),(18,33,74),(25,70,62),(26,71,63),(27,72,61),(28,57,79),(29,55,80),(30,56,81),(43,49,65),(44,50,66),(45,51,64)], [(1,73,38),(2,74,39),(3,75,37),(4,19,18),(5,20,16),(6,21,17),(7,78,11),(8,76,12),(9,77,10),(13,23,35),(14,24,36),(15,22,34),(25,50,81),(26,51,79),(27,49,80),(28,71,64),(29,72,65),(30,70,66),(31,54,60),(32,52,58),(33,53,59),(40,68,48),(41,69,46),(42,67,47),(43,55,61),(44,56,62),(45,57,63)], [(1,14,65),(2,15,66),(3,13,64),(4,40,56),(5,41,57),(6,42,55),(7,80,58),(8,81,59),(9,79,60),(10,51,54),(11,49,52),(12,50,53),(16,46,45),(17,47,43),(18,48,44),(19,68,62),(20,69,63),(21,67,61),(22,30,74),(23,28,75),(24,29,73),(25,33,76),(26,31,77),(27,32,78),(34,70,39),(35,71,37),(36,72,38)], [(1,22,45),(2,23,43),(3,24,44),(4,46,49),(5,47,50),(6,48,51),(7,70,60),(8,71,58),(9,72,59),(10,29,53),(11,30,54),(12,28,52),(13,61,39),(14,62,37),(15,63,38),(16,67,25),(17,68,26),(18,69,27),(19,41,80),(20,42,81),(21,40,79),(31,78,66),(32,76,64),(33,77,65),(34,57,73),(35,55,74),(36,56,75)], [(2,3),(4,16),(5,18),(6,17),(7,72),(8,71),(9,70),(10,66),(11,65),(12,64),(13,56),(14,55),(15,57),(19,20),(22,45),(23,44),(24,43),(25,46),(26,48),(27,47),(28,76),(29,78),(30,77),(31,53),(32,52),(33,54),(34,63),(35,62),(36,61),(37,74),(38,73),(39,75),(40,79),(41,81),(42,80),(49,67),(50,69),(51,68),(59,60)]])
Matrix representation of C3⋊(He3⋊S3) ►in GL8(𝔽19)
| 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 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 18 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 18 |
| 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 | 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 |
| 18 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 1 | 1 | 15 | 1 | 15 |
| 0 | 0 | 18 | 4 | 4 | 16 | 4 | 16 |
| 0 | 0 | 3 | 1 | 15 | 3 | 3 | 1 |
| 0 | 0 | 18 | 4 | 16 | 18 | 18 | 4 |
| 0 | 0 | 15 | 3 | 15 | 3 | 1 | 15 |
| 0 | 0 | 16 | 18 | 16 | 18 | 4 | 16 |
| 0 | 18 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 18 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 3 | 15 | 3 | 1 | 15 |
| 0 | 0 | 16 | 18 | 16 | 18 | 4 | 16 |
| 0 | 0 | 15 | 3 | 3 | 1 | 3 | 1 |
| 0 | 0 | 16 | 18 | 18 | 4 | 18 | 4 |
| 0 | 0 | 1 | 15 | 3 | 1 | 1 | 15 |
| 0 | 0 | 4 | 16 | 18 | 4 | 4 | 16 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,GF(19))| [0,1,0,0,0,0,0,0,18,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,18,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,18,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,18,18],[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],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,18,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,18,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,18,18],[18,18,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,3,18,3,18,15,16,0,0,1,4,1,4,3,18,0,0,1,4,15,16,15,16,0,0,15,16,3,18,3,18,0,0,1,4,3,18,1,4,0,0,15,16,1,4,15,16],[0,1,0,0,0,0,0,0,18,18,0,0,0,0,0,0,0,0,15,16,15,16,1,4,0,0,3,18,3,18,15,16,0,0,15,16,3,18,3,18,0,0,3,18,1,4,1,4,0,0,1,4,3,18,1,4,0,0,15,16,1,4,15,16],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
C3⋊(He3⋊S3) in GAP, Magma, Sage, TeX
C_3\rtimes ({\rm He}_3\rtimes S_3) % in TeX
G:=Group("C3:(He3:S3)"); // GroupNames label
G:=SmallGroup(486,187);
// by ID
G=gap.SmallGroup(486,187);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,49,218,867,303,1096,652,11669]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^3=e^3=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f=a^-1,e*b*e^-1=b*c=c*b,d*b*d^-1=b*c^-1,b*f=f*b,c*d=d*c,c*e=e*c,f*c*f=c^-1,e*d*e^-1=b^-1*c^-1*d,f*d*f=b*c^-1*d^-1,f*e*f=e^-1>;
// generators/relations
Export