G = C3.(He3⋊S3) order 486 = 2·35
3rd non-split extension by C3 of He3⋊S3 acting via He3⋊S3/He3⋊C3=C2
non-abelian, supersoluble, monomial
Aliases: C32⋊C9.9S3, C33.6(C3⋊S3), C3.3(He3⋊S3), C32.29He3⋊3C2, C3.4(He3.3S3), C32.18(He3⋊C2), C3.3(3- 1+2.S3), SmallGroup(486,48)
Series: Derived ►Chief ►Lower central ►Upper central
| C32.29He3 — C3.(He3⋊S3) |
Generators and relations for C3.(He3⋊S3)
G = < a,b,c,d,e,f | a3=b3=c3=f2=1, d3=e3=a, ab=ba, ac=ca, ad=da, ae=ea, faf=a-1, bc=cb, dbd-1=abc-1, ebe-1=abc, fbf=a-1b, cd=dc, ce=ec, fcf=c-1, ede-1=b-1c-1d, fdf=abc-1d2, fef=a-1e2 >
81C2
9C3
27S3
27S3
27S3
27S3
81C6
3C32
3C32
3C32
3C32
9C9
9C9
9C9
9C9
27D9
27D9
27C3×S3
27C3×S3
27D9
27C3×S3
27C3×S3
27D9
Character table of C3.(He3⋊S3)
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 6A | 6B | 9A | 9B | 9C | 9D | 9E | 9F | 9G | 9H | 9I | 9J | 9K | 9L | |
| size | 1 | 81 | 2 | 2 | 2 | 2 | 9 | 9 | 81 | 81 | 18 | 18 | 18 | 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 | trivial |
| ρ2 | 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 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ4 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | orthogonal lifted from S3 |
| ρ5 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 3 | 1 | 3 | 3 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | ζ3 | ζ32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3⋊C2 |
| ρ8 | 3 | -1 | 3 | 3 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3⋊C2 |
| ρ9 | 3 | 1 | 3 | 3 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | ζ32 | ζ3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3⋊C2 |
| ρ10 | 3 | -1 | 3 | 3 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3⋊C2 |
| ρ11 | 6 | 0 | 6 | -3 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | 0 | 0 | 0 | orthogonal lifted from He3⋊S3 |
| ρ12 | 6 | 0 | -3 | 6 | -3 | -3 | 0 | 0 | 0 | 0 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 3- 1+2.S3 |
| ρ13 | 6 | 0 | -3 | -3 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ97-ζ94+2ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from He3.3S3 |
| ρ14 | 6 | 0 | -3 | -3 | -3 | 6 | 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 | orthogonal lifted from 3- 1+2.S3 |
| ρ15 | 6 | 0 | -3 | -3 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ97-ζ94+2ζ92 | 2ζ95+ζ94+ζ92-ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from He3.3S3 |
| ρ16 | 6 | 0 | 6 | -3 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | 0 | 0 | 0 | orthogonal lifted from He3⋊S3 |
| ρ17 | 6 | 0 | -3 | 6 | -3 | -3 | 0 | 0 | 0 | 0 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 3- 1+2.S3 |
| ρ18 | 6 | 0 | -3 | -3 | -3 | 6 | 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 | orthogonal lifted from 3- 1+2.S3 |
| ρ19 | 6 | 0 | -3 | 6 | -3 | -3 | 0 | 0 | 0 | 0 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 3- 1+2.S3 |
| ρ20 | 6 | 0 | -3 | -3 | -3 | 6 | 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 | orthogonal lifted from 3- 1+2.S3 |
| ρ21 | 6 | 0 | 6 | -3 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 0 | 0 | 0 | orthogonal lifted from He3⋊S3 |
| ρ22 | 6 | 0 | -3 | -3 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ98+ζ97-ζ94+2ζ92 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ94-ζ92+2ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from He3.3S3 |
►On 81 points
Generators in S81
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)(73 76 79)(74 77 80)(75 78 81)
(1 7 4)(2 43 36)(3 31 41)(5 37 30)(6 34 44)(8 40 33)(9 28 38)(10 81 60)(12 62 74)(13 75 63)(15 56 77)(16 78 57)(18 59 80)(19 69 54)(20 52 64)(21 24 27)(22 72 48)(23 46 67)(25 66 51)(26 49 70)(29 35 32)(39 45 42)(47 50 53)(65 68 71)
(1 35 42)(2 36 43)(3 28 44)(4 29 45)(5 30 37)(6 31 38)(7 32 39)(8 33 40)(9 34 41)(10 75 57)(11 76 58)(12 77 59)(13 78 60)(14 79 61)(15 80 62)(16 81 63)(17 73 55)(18 74 56)(19 51 72)(20 52 64)(21 53 65)(22 54 66)(23 46 67)(24 47 68)(25 48 69)(26 49 70)(27 50 71)
(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 26 59 4 20 62 7 23 56)(2 47 57 5 50 60 8 53 63)(3 48 14 6 51 17 9 54 11)(10 30 71 13 33 65 16 36 68)(12 29 52 15 32 46 18 35 49)(19 55 41 22 58 44 25 61 38)(21 81 43 24 75 37 27 78 40)(28 69 79 31 72 73 34 66 76)(39 67 74 42 70 77 45 64 80)
(2 44)(3 43)(4 7)(5 41)(6 40)(8 38)(9 37)(10 66)(11 24)(12 67)(13 72)(14 21)(15 64)(16 69)(17 27)(18 70)(19 60)(20 62)(22 57)(23 59)(25 63)(26 56)(28 36)(29 39)(30 34)(31 33)(32 45)(35 42)(46 77)(47 58)(48 81)(49 74)(50 55)(51 78)(52 80)(53 61)(54 75)(65 79)(68 76)(71 73)
G:=sub<Sym(81)| (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81), (1,7,4)(2,43,36)(3,31,41)(5,37,30)(6,34,44)(8,40,33)(9,28,38)(10,81,60)(12,62,74)(13,75,63)(15,56,77)(16,78,57)(18,59,80)(19,69,54)(20,52,64)(21,24,27)(22,72,48)(23,46,67)(25,66,51)(26,49,70)(29,35,32)(39,45,42)(47,50,53)(65,68,71), (1,35,42)(2,36,43)(3,28,44)(4,29,45)(5,30,37)(6,31,38)(7,32,39)(8,33,40)(9,34,41)(10,75,57)(11,76,58)(12,77,59)(13,78,60)(14,79,61)(15,80,62)(16,81,63)(17,73,55)(18,74,56)(19,51,72)(20,52,64)(21,53,65)(22,54,66)(23,46,67)(24,47,68)(25,48,69)(26,49,70)(27,50,71), (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), (1,26,59,4,20,62,7,23,56)(2,47,57,5,50,60,8,53,63)(3,48,14,6,51,17,9,54,11)(10,30,71,13,33,65,16,36,68)(12,29,52,15,32,46,18,35,49)(19,55,41,22,58,44,25,61,38)(21,81,43,24,75,37,27,78,40)(28,69,79,31,72,73,34,66,76)(39,67,74,42,70,77,45,64,80), (2,44)(3,43)(4,7)(5,41)(6,40)(8,38)(9,37)(10,66)(11,24)(12,67)(13,72)(14,21)(15,64)(16,69)(17,27)(18,70)(19,60)(20,62)(22,57)(23,59)(25,63)(26,56)(28,36)(29,39)(30,34)(31,33)(32,45)(35,42)(46,77)(47,58)(48,81)(49,74)(50,55)(51,78)(52,80)(53,61)(54,75)(65,79)(68,76)(71,73)>;
G:=Group( (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81), (1,7,4)(2,43,36)(3,31,41)(5,37,30)(6,34,44)(8,40,33)(9,28,38)(10,81,60)(12,62,74)(13,75,63)(15,56,77)(16,78,57)(18,59,80)(19,69,54)(20,52,64)(21,24,27)(22,72,48)(23,46,67)(25,66,51)(26,49,70)(29,35,32)(39,45,42)(47,50,53)(65,68,71), (1,35,42)(2,36,43)(3,28,44)(4,29,45)(5,30,37)(6,31,38)(7,32,39)(8,33,40)(9,34,41)(10,75,57)(11,76,58)(12,77,59)(13,78,60)(14,79,61)(15,80,62)(16,81,63)(17,73,55)(18,74,56)(19,51,72)(20,52,64)(21,53,65)(22,54,66)(23,46,67)(24,47,68)(25,48,69)(26,49,70)(27,50,71), (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), (1,26,59,4,20,62,7,23,56)(2,47,57,5,50,60,8,53,63)(3,48,14,6,51,17,9,54,11)(10,30,71,13,33,65,16,36,68)(12,29,52,15,32,46,18,35,49)(19,55,41,22,58,44,25,61,38)(21,81,43,24,75,37,27,78,40)(28,69,79,31,72,73,34,66,76)(39,67,74,42,70,77,45,64,80), (2,44)(3,43)(4,7)(5,41)(6,40)(8,38)(9,37)(10,66)(11,24)(12,67)(13,72)(14,21)(15,64)(16,69)(17,27)(18,70)(19,60)(20,62)(22,57)(23,59)(25,63)(26,56)(28,36)(29,39)(30,34)(31,33)(32,45)(35,42)(46,77)(47,58)(48,81)(49,74)(50,55)(51,78)(52,80)(53,61)(54,75)(65,79)(68,76)(71,73) );
G=PermutationGroup([[(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54),(55,58,61),(56,59,62),(57,60,63),(64,67,70),(65,68,71),(66,69,72),(73,76,79),(74,77,80),(75,78,81)], [(1,7,4),(2,43,36),(3,31,41),(5,37,30),(6,34,44),(8,40,33),(9,28,38),(10,81,60),(12,62,74),(13,75,63),(15,56,77),(16,78,57),(18,59,80),(19,69,54),(20,52,64),(21,24,27),(22,72,48),(23,46,67),(25,66,51),(26,49,70),(29,35,32),(39,45,42),(47,50,53),(65,68,71)], [(1,35,42),(2,36,43),(3,28,44),(4,29,45),(5,30,37),(6,31,38),(7,32,39),(8,33,40),(9,34,41),(10,75,57),(11,76,58),(12,77,59),(13,78,60),(14,79,61),(15,80,62),(16,81,63),(17,73,55),(18,74,56),(19,51,72),(20,52,64),(21,53,65),(22,54,66),(23,46,67),(24,47,68),(25,48,69),(26,49,70),(27,50,71)], [(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)], [(1,26,59,4,20,62,7,23,56),(2,47,57,5,50,60,8,53,63),(3,48,14,6,51,17,9,54,11),(10,30,71,13,33,65,16,36,68),(12,29,52,15,32,46,18,35,49),(19,55,41,22,58,44,25,61,38),(21,81,43,24,75,37,27,78,40),(28,69,79,31,72,73,34,66,76),(39,67,74,42,70,77,45,64,80)], [(2,44),(3,43),(4,7),(5,41),(6,40),(8,38),(9,37),(10,66),(11,24),(12,67),(13,72),(14,21),(15,64),(16,69),(17,27),(18,70),(19,60),(20,62),(22,57),(23,59),(25,63),(26,56),(28,36),(29,39),(30,34),(31,33),(32,45),(35,42),(46,77),(47,58),(48,81),(49,74),(50,55),(51,78),(52,80),(53,61),(54,75),(65,79),(68,76),(71,73)]])
Matrix representation of C3.(He3⋊S3) ►in GL12(𝔽19)
| 1 | 0 | 0 | 0 | 0 | 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 | 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 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 | 18 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 | 18 | 18 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | 10 | 18 | 18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 18 | 18 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 18 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 18 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 | 18 | 18 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 1 | 0 |
| 0 | 18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | 0 | 18 | 18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 18 | 18 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 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 | 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 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 13 | 13 | 7 | 10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 16 | 7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 13 | 13 | 7 | 6 | 2 | 14 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | 10 | 7 | 6 | 5 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 6 | 16 | 11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 1 | 16 | 11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 6 | 6 | 0 | 0 | 9 | 16 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 12 | 9 |
| 0 | 0 | 0 | 0 | 0 | 0 | 14 | 7 | 0 | 0 | 17 | 13 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 14 | 0 | 0 | 17 | 13 |
| 0 | 0 | 0 | 0 | 0 | 0 | 8 | 8 | 2 | 14 | 17 | 13 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 4 | 5 | 7 | 17 | 13 |
| 0 | 0 | 0 | 0 | 18 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 17 | 18 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 8 | 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 8 | 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 13 | 13 | 1 | 0 | 7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 13 | 13 | 0 | 1 | 7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 16 | 0 | 0 | 18 | 17 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 18 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 10 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 9 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 18 | 18 | 0 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 4 | 1 | 0 | 0 | 3 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 5 | 5 | 0 | 0 | 18 | 18 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 7 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 17 | 17 | 18 | 18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 1 | 1 | 0 | 0 |
G:=sub<GL(12,GF(19))| [1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,16,0,16,0,0,0,0,0,0,0,18,18,0,3,0,3,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0],[1,0,10,0,0,12,0,0,0,0,0,0,0,1,10,0,0,12,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,18,18,3,3,0,3,0,0,0,0,0,0,1,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0],[0,1,10,0,12,0,0,0,0,0,0,0,18,18,0,9,0,7,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[13,12,13,10,1,8,0,0,0,0,0,0,13,12,13,10,6,1,0,0,0,0,0,0,7,16,7,7,16,16,0,0,0,0,0,0,10,7,6,6,11,11,0,0,0,0,0,0,0,0,2,5,0,0,0,0,0,0,0,0,0,0,14,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,2,14,12,8,4,0,0,0,0,0,0,6,2,7,14,8,4,0,0,0,0,0,0,0,0,0,0,2,5,0,0,0,0,0,0,0,0,0,0,14,7,0,0,0,0,0,0,9,12,17,17,17,17,0,0,0,0,0,0,16,9,13,13,13,13],[0,12,8,9,13,13,0,0,0,0,0,0,0,12,8,8,13,13,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,18,17,9,9,7,7,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,9,9,1,4,0,0,0,0,0,0,16,0,10,9,1,4,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,17,18,3,3,3,3],[0,1,12,5,7,17,0,0,0,0,0,0,1,0,12,5,7,17,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,16,0,16,0,0,0,0,0,0,0,0,18,0,3,0,3,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0] >;
C3.(He3⋊S3) in GAP, Magma, Sage, TeX
C_3.({\rm He}_3\rtimes S_3) % in TeX
G:=Group("C3.(He3:S3)"); // GroupNames label
G:=SmallGroup(486,48);
// by ID
G=gap.SmallGroup(486,48);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,697,655,1190,224,338,6915,2817,735,3244,11669]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=f^2=1,d^3=e^3=a,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f=a^-1,b*c=c*b,d*b*d^-1=a*b*c^-1,e*b*e^-1=a*b*c,f*b*f=a^-1*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=a*b*c^-1*d^2,f*e*f=a^-1*e^2>;
// generators/relations
Export
Subgroup lattice of C3.(He3⋊S3) in TeX
Character table of C3.(He3⋊S3) in TeX