direct product, metabelian, supersoluble, monomial
Aliases: S3×He3, C33⋊3C6, C3⋊(C2×He3), (S3×C32)⋊C3, (C3×He3)⋊2C2, C32⋊4(C3×S3), C3.5(S3×C32), C32.7(C3×C6), (C3×S3).2C32, SmallGroup(162,35)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×He3
G = < a,b,c,d,e | a3=b2=c3=d3=e3=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >
Subgroups: 180 in 61 conjugacy classes, 21 normal (9 characteristic)
C1, C2, C3, C3, S3, C6, C32, C32, C32, C3×S3, C3×S3, C3×C6, He3, He3, C33, C2×He3, S3×C32, C3×He3, S3×He3
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, He3, C2×He3, S3×C32, S3×He3
►On 18 points - transitive group 18T77
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 11)(2 10)(3 12)(4 14)(5 13)(6 15)(7 17)(8 16)(9 18)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)
(1 2 3)(4 5 6)(7 8 9)(10 12 11)(13 15 14)(16 18 17)
(4 6 5)(7 8 9)(13 14 15)(16 18 17)
G:=sub<Sym(18)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,11)(2,10)(3,12)(4,14)(5,13)(6,15)(7,17)(8,16)(9,18), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15), (1,2,3)(4,5,6)(7,8,9)(10,12,11)(13,15,14)(16,18,17), (4,6,5)(7,8,9)(13,14,15)(16,18,17)>;
G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,11)(2,10)(3,12)(4,14)(5,13)(6,15)(7,17)(8,16)(9,18), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15), (1,2,3)(4,5,6)(7,8,9)(10,12,11)(13,15,14)(16,18,17), (4,6,5)(7,8,9)(13,14,15)(16,18,17) );
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,11),(2,10),(3,12),(4,14),(5,13),(6,15),(7,17),(8,16),(9,18)], [(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15)], [(1,2,3),(4,5,6),(7,8,9),(10,12,11),(13,15,14),(16,18,17)], [(4,6,5),(7,8,9),(13,14,15),(16,18,17)]])
G:=TransitiveGroup(18,77);
►On 27 points - transitive group 27T73
Generators in S27
(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 3)(5 6)(8 9)(11 12)(14 15)(17 18)(20 21)(23 24)(26 27)
(10 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 27)
(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)
(1 19 10)(2 20 11)(3 21 12)(4 22 13)(5 23 14)(6 24 15)(7 25 16)(8 26 17)(9 27 18)
G:=sub<Sym(27)| (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), (2,3)(5,6)(8,9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27), (10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27), (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), (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)>;
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), (2,3)(5,6)(8,9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27), (10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27), (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), (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18) );
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)], [(2,3),(5,6),(8,9),(11,12),(14,15),(17,18),(20,21),(23,24),(26,27)], [(10,16,13),(11,17,14),(12,18,15),(19,22,25),(20,23,26),(21,24,27)], [(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)], [(1,19,10),(2,20,11),(3,21,12),(4,22,13),(5,23,14),(6,24,15),(7,25,16),(8,26,17),(9,27,18)]])
G:=TransitiveGroup(27,73);
S3×He3 is a maximal subgroup of
C34⋊C6 D9⋊He3 C3≀C3⋊C6 3+ 1+4⋊2C2
S3×He3 is a maximal quotient of C34⋊C6 D9⋊He3
33 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | ··· | 3M | 3N | ··· | 3U | 6A | 6B | 6C | ··· | 6J |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 3 | ··· | 3 | 6 | 6 | 6 | ··· | 6 |
| size | 1 | 3 | 1 | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 3 | 3 | 9 | ··· | 9 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 6 |
| type | + | + | + | ||||||
| image | C1 | C2 | C3 | C6 | S3 | C3×S3 | He3 | C2×He3 | S3×He3 |
| kernel | S3×He3 | C3×He3 | S3×C32 | C33 | He3 | C32 | S3 | C3 | C1 |
| # reps | 1 | 1 | 8 | 8 | 1 | 8 | 2 | 2 | 2 |
Matrix representation of S3×He3 ►in GL5(𝔽7)
| 0 | 1 | 0 | 0 | 0 |
| 6 | 6 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 2 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 |
G:=sub<GL(5,GF(7))| [0,6,0,0,0,1,6,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[2,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,4],[1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,5,0,0,1,0,0] >;
S3×He3 in GAP, Magma, Sage, TeX
S_3\times {\rm He}_3 % in TeX
G:=Group("S3xHe3"); // GroupNames label
G:=SmallGroup(162,35);
// by ID
G=gap.SmallGroup(162,35);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,187,2704]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^3=d^3=e^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c*d^-1,d*e=e*d>;
// generators/relations