metabelian, supersoluble, monomial
Aliases: He3.2S3, C9⋊S3⋊3C3, (C3×C9)⋊3C6, He3⋊C3⋊2C2, C32.8(C3×S3), C3.4(C32⋊C6), SmallGroup(162,15)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C9 — He3.2S3 |
Generators and relations for He3.2S3
G = < a,b,c,d,e | a3=b3=c3=e2=1, d3=b, ab=ba, cac-1=ab-1, ad=da, eae=a-1, bc=cb, bd=db, ebe=b-1, dcd-1=a-1bc, ce=ec, ede=b-1d2 >
Character table of He3.2S3
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 6A | 6B | 9A | 9B | 9C | |
| size | 1 | 27 | 2 | 6 | 9 | 9 | 18 | 18 | 27 | 27 | 6 | 6 | 6 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | 1 | 1 | linear of order 6 |
| ρ4 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | -1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | 1 | 1 | linear of order 6 |
| ρ7 | 2 | 0 | 2 | 2 | 2 | 2 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | 2 | 2 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | 0 | 0 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ9 | 2 | 0 | 2 | 2 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | 0 | 0 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ10 | 6 | 0 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C6 |
| ρ11 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | orthogonal faithful |
| ρ12 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | orthogonal faithful |
| ρ13 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | orthogonal faithful |
►On 27 points - transitive group 27T38
Generators in S27
(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)
(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)
(1 25 13)(2 20 14)(3 24 15)(4 19 16)(5 23 17)(6 27 18)(7 22 10)(8 26 11)(9 21 12)
(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)
(2 9)(3 8)(4 7)(5 6)(10 16)(11 15)(12 14)(17 18)(19 22)(20 21)(23 27)(24 26)
G:=sub<Sym(27)| (10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24), (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), (1,25,13)(2,20,14)(3,24,15)(4,19,16)(5,23,17)(6,27,18)(7,22,10)(8,26,11)(9,21,12), (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), (2,9)(3,8)(4,7)(5,6)(10,16)(11,15)(12,14)(17,18)(19,22)(20,21)(23,27)(24,26)>;
G:=Group( (10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24), (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), (1,25,13)(2,20,14)(3,24,15)(4,19,16)(5,23,17)(6,27,18)(7,22,10)(8,26,11)(9,21,12), (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), (2,9)(3,8)(4,7)(5,6)(10,16)(11,15)(12,14)(17,18)(19,22)(20,21)(23,27)(24,26) );
G=PermutationGroup([[(10,13,16),(11,14,17),(12,15,18),(19,25,22),(20,26,23),(21,27,24)], [(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)], [(1,25,13),(2,20,14),(3,24,15),(4,19,16),(5,23,17),(6,27,18),(7,22,10),(8,26,11),(9,21,12)], [(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)], [(2,9),(3,8),(4,7),(5,6),(10,16),(11,15),(12,14),(17,18),(19,22),(20,21),(23,27),(24,26)]])
G:=TransitiveGroup(27,38);
►On 27 points - transitive group 27T64
Generators in S27
(1 14 19)(2 15 20)(3 16 21)(4 17 22)(5 18 23)(6 10 24)(7 11 25)(8 12 26)(9 13 27)
(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)
(2 20 12)(3 16 24)(5 23 15)(6 10 27)(8 26 18)(9 13 21)(11 14 17)(19 25 22)
(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)
(2 9)(3 8)(4 7)(5 6)(10 23)(11 22)(12 21)(13 20)(14 19)(15 27)(16 26)(17 25)(18 24)
G:=sub<Sym(27)| (1,14,19)(2,15,20)(3,16,21)(4,17,22)(5,18,23)(6,10,24)(7,11,25)(8,12,26)(9,13,27), (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), (2,20,12)(3,16,24)(5,23,15)(6,10,27)(8,26,18)(9,13,21)(11,14,17)(19,25,22), (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), (2,9)(3,8)(4,7)(5,6)(10,23)(11,22)(12,21)(13,20)(14,19)(15,27)(16,26)(17,25)(18,24)>;
G:=Group( (1,14,19)(2,15,20)(3,16,21)(4,17,22)(5,18,23)(6,10,24)(7,11,25)(8,12,26)(9,13,27), (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), (2,20,12)(3,16,24)(5,23,15)(6,10,27)(8,26,18)(9,13,21)(11,14,17)(19,25,22), (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), (2,9)(3,8)(4,7)(5,6)(10,23)(11,22)(12,21)(13,20)(14,19)(15,27)(16,26)(17,25)(18,24) );
G=PermutationGroup([[(1,14,19),(2,15,20),(3,16,21),(4,17,22),(5,18,23),(6,10,24),(7,11,25),(8,12,26),(9,13,27)], [(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)], [(2,20,12),(3,16,24),(5,23,15),(6,10,27),(8,26,18),(9,13,21),(11,14,17),(19,25,22)], [(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)], [(2,9),(3,8),(4,7),(5,6),(10,23),(11,22),(12,21),(13,20),(14,19),(15,27),(16,26),(17,25),(18,24)]])
G:=TransitiveGroup(27,64);
He3.2S3 is a maximal subgroup of
He3.2D6 C92⋊C6 C92⋊2C6 He3.(C3×S3) He3⋊C3⋊3S3 C3≀C3.S3
He3.2S3 is a maximal quotient of
He3.2Dic3 C32⋊C9.S3 C3.3C3≀S3 C33.(C3×S3) C9⋊S3⋊3C9 He3⋊D9 C92⋊C6 C92⋊2C6 He3⋊C3⋊3S3
Matrix representation of He3.2S3 ►in GL6(𝔽19)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 14 | 9 | 18 | 18 | 0 | 0 |
| 5 | 2 | 0 | 0 | 18 | 18 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 18 | 1 | 0 | 0 | 0 | 0 |
| 18 | 0 | 0 | 0 | 0 | 0 |
| 13 | 8 | 0 | 1 | 0 | 0 |
| 8 | 17 | 18 | 18 | 0 | 0 |
| 15 | 1 | 0 | 0 | 0 | 1 |
| 1 | 3 | 0 | 0 | 18 | 18 |
| 0 | 0 | 18 | 1 | 0 | 0 |
| 14 | 9 | 17 | 18 | 0 | 0 |
| 15 | 1 | 2 | 8 | 1 | 0 |
| 15 | 1 | 2 | 8 | 0 | 1 |
| 9 | 12 | 16 | 1 | 0 | 0 |
| 10 | 12 | 16 | 1 | 0 | 0 |
| 5 | 12 | 0 | 0 | 0 | 0 |
| 7 | 17 | 0 | 0 | 0 | 0 |
| 4 | 1 | 17 | 12 | 0 | 0 |
| 1 | 14 | 7 | 5 | 0 | 0 |
| 18 | 8 | 0 | 0 | 14 | 2 |
| 9 | 12 | 0 | 0 | 17 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 11 | 8 | 1 | 0 | 0 | 0 |
| 6 | 17 | 18 | 18 | 0 | 0 |
| 18 | 1 | 0 | 0 | 1 | 0 |
| 4 | 3 | 0 | 0 | 18 | 18 |
G:=sub<GL(6,GF(19))| [1,0,0,14,5,0,0,1,0,9,2,0,0,0,0,18,0,0,0,0,1,18,0,0,0,0,0,0,18,1,0,0,0,0,18,0],[18,18,13,8,15,1,1,0,8,17,1,3,0,0,0,18,0,0,0,0,1,18,0,0,0,0,0,0,0,18,0,0,0,0,1,18],[0,14,15,15,9,10,0,9,1,1,12,12,18,17,2,2,16,16,1,18,8,8,1,1,0,0,1,0,0,0,0,0,0,1,0,0],[5,7,4,1,18,9,12,17,1,14,8,12,0,0,17,7,0,0,0,0,12,5,0,0,0,0,0,0,14,17,0,0,0,0,2,12],[0,1,11,6,18,4,1,0,8,17,1,3,0,0,1,18,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,18] >;
He3.2S3 in GAP, Magma, Sage, TeX
{\rm He}_3._2S_3 % in TeX
G:=Group("He3.2S3"); // GroupNames label
G:=SmallGroup(162,15);
// by ID
G=gap.SmallGroup(162,15);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,992,187,282,723,728,2704]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=e^2=1,d^3=b,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=a^-1*b*c,c*e=e*c,e*d*e=b^-1*d^2>;
// generators/relations
Export
Subgroup lattice of He3.2S3 in TeX
Character table of He3.2S3 in TeX