non-abelian, supersoluble, monomial
Aliases: C3≀S3⋊3C3, C3≀C3.2C6, C9○He3⋊2C6, (C32×C9)⋊24S3, He3.3(C3×C6), C9.He3⋊3C2, He3.C3⋊10C6, C9.9(C32⋊C6), C33.37(C3×S3), He3⋊C3⋊10C6, He3.4C6⋊1C3, He3.2C6⋊3C3, He3.C6⋊3C3, C32.4(S3×C32), He3⋊C2.2C32, (C3×C9).25(C3×S3), C3.18(C3×C32⋊C6), SmallGroup(486,125)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — C3≀S3⋊3C3 |
Generators and relations for C3≀S3⋊3C3
G = < a,b,c,d,e | a3=b3=c3=d6=e3=1, eae-1=ab=ba, cac-1=ab-1, dad-1=a-1b, bc=cb, bd=db, be=eb, dcd-1=a-1c-1, ece-1=ab-1c, ede-1=bcd >
Subgroups: 306 in 77 conjugacy classes, 21 normal (all characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C32, C32, C18, C3×S3, C3×C6, C3×C9, C3×C9, He3, He3, 3- 1+2, C33, S3×C9, He3⋊C2, C3×C18, S3×C32, C3≀C3, C3≀C3, He3.C3, He3.C3, He3⋊C3, C3.He3, C32×C9, C9○He3, C9○He3, C3≀S3, He3.C6, He3.2C6, S3×C3×C9, He3.4C6, C9.He3, C3≀S3⋊3C3
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, C32⋊C6, S3×C32, C3×C32⋊C6, C3≀S3⋊3C3
►On 27 points - transitive group 27T149
(1 3 9)(2 8 6)(4 7 5)(11 13 15)(17 19 21)(22 24 26)
(1 9 3)(2 6 8)(4 5 7)(10 12 14)(11 13 15)(16 18 20)(17 19 21)(22 24 26)(23 25 27)
(1 26 23)(2 17 20)(3 24 27)(4 15 12)(5 11 14)(6 19 16)(7 13 10)(8 21 18)(9 22 25)
(10 11 12 13 14 15)(16 17 18 19 20 21)(22 23 24 25 26 27)
(1 14 19)(2 23 15)(3 12 17)(4 16 26)(5 18 22)(6 25 11)(7 20 24)(8 27 13)(9 10 21)
G:=sub<Sym(27)| (1,3,9)(2,8,6)(4,7,5)(11,13,15)(17,19,21)(22,24,26), (1,9,3)(2,6,8)(4,5,7)(10,12,14)(11,13,15)(16,18,20)(17,19,21)(22,24,26)(23,25,27), (1,26,23)(2,17,20)(3,24,27)(4,15,12)(5,11,14)(6,19,16)(7,13,10)(8,21,18)(9,22,25), (10,11,12,13,14,15)(16,17,18,19,20,21)(22,23,24,25,26,27), (1,14,19)(2,23,15)(3,12,17)(4,16,26)(5,18,22)(6,25,11)(7,20,24)(8,27,13)(9,10,21)>;
G:=Group( (1,3,9)(2,8,6)(4,7,5)(11,13,15)(17,19,21)(22,24,26), (1,9,3)(2,6,8)(4,5,7)(10,12,14)(11,13,15)(16,18,20)(17,19,21)(22,24,26)(23,25,27), (1,26,23)(2,17,20)(3,24,27)(4,15,12)(5,11,14)(6,19,16)(7,13,10)(8,21,18)(9,22,25), (10,11,12,13,14,15)(16,17,18,19,20,21)(22,23,24,25,26,27), (1,14,19)(2,23,15)(3,12,17)(4,16,26)(5,18,22)(6,25,11)(7,20,24)(8,27,13)(9,10,21) );
G=PermutationGroup([[(1,3,9),(2,8,6),(4,7,5),(11,13,15),(17,19,21),(22,24,26)], [(1,9,3),(2,6,8),(4,5,7),(10,12,14),(11,13,15),(16,18,20),(17,19,21),(22,24,26),(23,25,27)], [(1,26,23),(2,17,20),(3,24,27),(4,15,12),(5,11,14),(6,19,16),(7,13,10),(8,21,18),(9,22,25)], [(10,11,12,13,14,15),(16,17,18,19,20,21),(22,23,24,25,26,27)], [(1,14,19),(2,23,15),(3,12,17),(4,16,26),(5,18,22),(6,25,11),(7,20,24),(8,27,13),(9,10,21)]])
G:=TransitiveGroup(27,149);
66 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3H | 3I | 3J | 3K | 3L | 6A | ··· | 6H | 9A | ··· | 9F | 9G | ··· | 9R | 9S | 9T | 9U | ··· | 9Z | 18A | ··· | 18R |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 9 | 9 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 9 | 1 | 1 | 3 | ··· | 3 | 6 | 18 | 18 | 18 | 9 | ··· | 9 | 1 | ··· | 1 | 3 | ··· | 3 | 6 | 6 | 18 | ··· | 18 | 9 | ··· | 9 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 6 | 6 |
| type | + | + | + | + | ||||||||||||
| image | C1 | C2 | C3 | C3 | C3 | C3 | C6 | C6 | C6 | C6 | S3 | C3×S3 | C3×S3 | C3≀S3⋊3C3 | C32⋊C6 | C3×C32⋊C6 |
| kernel | C3≀S3⋊3C3 | C9.He3 | C3≀S3 | He3.C6 | He3.2C6 | He3.4C6 | C3≀C3 | He3.C3 | He3⋊C3 | C9○He3 | C32×C9 | C3×C9 | C33 | C1 | C9 | C3 |
| # reps | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 6 | 2 | 36 | 1 | 2 |
Matrix representation of C3≀S3⋊3C3 ►in GL3(𝔽19) generated by
| 11 | 0 | 8 |
| 0 | 7 | 1 |
| 0 | 0 | 1 |
| 7 | 0 | 0 |
| 0 | 7 | 0 |
| 0 | 0 | 7 |
| 0 | 18 | 0 |
| 1 | 18 | 0 |
| 0 | 6 | 1 |
| 1 | 8 | 11 |
| 0 | 8 | 0 |
| 0 | 9 | 11 |
| 3 | 5 | 14 |
| 3 | 0 | 14 |
| 1 | 0 | 16 |
G:=sub<GL(3,GF(19))| [11,0,0,0,7,0,8,1,1],[7,0,0,0,7,0,0,0,7],[0,1,0,18,18,6,0,0,1],[1,0,0,8,8,9,11,0,11],[3,3,1,5,0,0,14,14,16] >;
C3≀S3⋊3C3 in GAP, Magma, Sage, TeX
C_3\wr S_3\rtimes_3C_3
% in TeX
G:=Group("C3wrS3:3C3"); // GroupNames label
G:=SmallGroup(486,125);
// by ID
G=gap.SmallGroup(486,125);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,500,867,873,8104,382]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^6=e^3=1,e*a*e^-1=a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1*b,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=a^-1*c^-1,e*c*e^-1=a*b^-1*c,e*d*e^-1=b*c*d>;
// generators/relations