non-abelian, supersoluble, monomial
Aliases: C92⋊2S3, He3⋊C3⋊S3, C92⋊2C3⋊3C2, C3.6(He3⋊S3), C32.1(He3⋊C2), (C3×C9).1(C3⋊S3), SmallGroup(486,61)
Series: Derived ►Chief ►Lower central ►Upper central
| C92⋊2C3 — C92⋊2S3 |
Generators and relations for C92⋊2S3
G = < a,b,c,d | a9=b9=c3=d2=1, ab=ba, cac-1=a7b-1, dad=a-1b-1, cbc-1=a3b, bd=db, dcd=c-1 >
27C2
3C3
27C3
27C3
27C3
9S3
27S3
27S3
27S3
27C6
3C9
3C9
6C9
9C32
9C32
9C32
3D9
27C3×S3
27C18
27C3×S3
27C3×S3
3He3
3He3
3He3
Permutation representations of C92⋊2S3
►On 27 points - transitive group 27T179
Generators in S27
(10 11 12 13 14 15 16 17 18)(19 20 21 22 23 24 25 26 27)
(1 9 2 5 8 6 4 3 7)(10 17 15 13 11 18 16 14 12)(19 27 26 25 24 23 22 21 20)
(1 27 15)(2 25 17)(3 20 13)(4 21 12)(5 24 18)(6 22 11)(7 19 14)(8 23 10)(9 26 16)
(1 27)(2 25)(3 20)(4 21)(5 24)(6 22)(7 19)(8 23)(9 26)
G:=sub<Sym(27)| (10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27), (1,9,2,5,8,6,4,3,7)(10,17,15,13,11,18,16,14,12)(19,27,26,25,24,23,22,21,20), (1,27,15)(2,25,17)(3,20,13)(4,21,12)(5,24,18)(6,22,11)(7,19,14)(8,23,10)(9,26,16), (1,27)(2,25)(3,20)(4,21)(5,24)(6,22)(7,19)(8,23)(9,26)>;
G:=Group( (10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27), (1,9,2,5,8,6,4,3,7)(10,17,15,13,11,18,16,14,12)(19,27,26,25,24,23,22,21,20), (1,27,15)(2,25,17)(3,20,13)(4,21,12)(5,24,18)(6,22,11)(7,19,14)(8,23,10)(9,26,16), (1,27)(2,25)(3,20)(4,21)(5,24)(6,22)(7,19)(8,23)(9,26) );
G=PermutationGroup([[(10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27)], [(1,9,2,5,8,6,4,3,7),(10,17,15,13,11,18,16,14,12),(19,27,26,25,24,23,22,21,20)], [(1,27,15),(2,25,17),(3,20,13),(4,21,12),(5,24,18),(6,22,11),(7,19,14),(8,23,10),(9,26,16)], [(1,27),(2,25),(3,20),(4,21),(5,24),(6,22),(7,19),(8,23),(9,26)]])
G:=TransitiveGroup(27,179);
31 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 6A | 6B | 9A | ··· | 9F | 9G | ··· | 9O | 18A | ··· | 18F |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 27 | 1 | 1 | 6 | 54 | 54 | 54 | 27 | 27 | 3 | ··· | 3 | 6 | ··· | 6 | 27 | ··· | 27 |
31 irreducible representations
| dim | 1 | 1 | 2 | 2 | 3 | 3 | 6 | 6 |
| type | + | + | + | + | + | |||
| image | C1 | C2 | S3 | S3 | He3⋊C2 | C92⋊2S3 | He3⋊S3 | C92⋊2S3 |
| kernel | C92⋊2S3 | C92⋊2C3 | C92 | He3⋊C3 | C32 | C1 | C3 | C1 |
| # reps | 1 | 1 | 1 | 3 | 4 | 12 | 3 | 6 |
Matrix representation of C92⋊2S3 ►in GL3(𝔽19) generated by
| 6 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 16 |
| 9 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(19))| [6,0,0,0,1,0,0,0,16],[9,0,0,0,6,0,0,0,6],[0,1,0,0,0,1,1,0,0],[1,0,0,0,0,1,0,1,0] >;
C92⋊2S3 in GAP, Magma, Sage, TeX
C_9^2\rtimes_2S_3
% in TeX
G:=Group("C9^2:2S3"); // GroupNames label
G:=SmallGroup(486,61);
// by ID
G=gap.SmallGroup(486,61);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,49,218,224,873,1167,453,8104,3250,1906]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^9=c^3=d^2=1,a*b=b*a,c*a*c^-1=a^7*b^-1,d*a*d=a^-1*b^-1,c*b*c^-1=a^3*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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