direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3×S3×D9, C32⋊6D18, C33.7D6, C9⋊S3⋊7C6, C9⋊4(S3×C6), C3⋊1(C6×D9), (S3×C9)⋊3C6, (C3×D9)⋊3C6, (C3×C9)⋊16D6, C32.10S32, (C32×D9)⋊1C2, C32.9(S3×C6), (C32×C9)⋊2C22, (S3×C32).2S3, (S3×C3×C9)⋊2C2, C3.1(C3×S32), (C3×C9⋊S3)⋊1C2, (C3×C9)⋊9(C2×C6), (C3×S3).1(C3×S3), SmallGroup(324,114)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C9 — C3×S3×D9 |
Generators and relations for C3×S3×D9
G = < a,b,c,d,e | a3=b3=c2=d9=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 424 in 86 conjugacy classes, 26 normal (all characteristic)
C1, C2, C3, C3, C22, S3, S3, C6, C9, C9, C32, C32, D6, C2×C6, D9, D9, C18, C3×S3, C3×S3, C3⋊S3, C3×C6, C3×C9, C3×C9, C33, D18, S32, S3×C6, C3×D9, C3×D9, S3×C9, S3×C9, C9⋊S3, C3×C18, S3×C32, S3×C32, C3×C3⋊S3, C32×C9, S3×D9, C6×D9, C3×S32, C32×D9, S3×C3×C9, C3×C9⋊S3, C3×S3×D9
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, D9, C3×S3, D18, S32, S3×C6, C3×D9, S3×D9, C6×D9, C3×S32, C3×S3×D9
►On 36 points
(1 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 27)(28 34 31)(29 35 32)(30 36 33)
(1 7 4)(2 8 5)(3 9 6)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 34 31)(29 35 32)(30 36 33)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 19)(7 20)(8 21)(9 22)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)
(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)
(1 12)(2 11)(3 10)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 36)(27 35)
G:=sub<Sym(36)| (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27)(28,34,31)(29,35,32)(30,36,33), (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,34,31)(29,35,32)(30,36,33), (1,23)(2,24)(3,25)(4,26)(5,27)(6,19)(7,20)(8,21)(9,22)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36), (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), (1,12)(2,11)(3,10)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,36)(27,35)>;
G:=Group( (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27)(28,34,31)(29,35,32)(30,36,33), (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,34,31)(29,35,32)(30,36,33), (1,23)(2,24)(3,25)(4,26)(5,27)(6,19)(7,20)(8,21)(9,22)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36), (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), (1,12)(2,11)(3,10)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,36)(27,35) );
G=PermutationGroup([[(1,4,7),(2,5,8),(3,6,9),(10,16,13),(11,17,14),(12,18,15),(19,22,25),(20,23,26),(21,24,27),(28,34,31),(29,35,32),(30,36,33)], [(1,7,4),(2,8,5),(3,9,6),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,34,31),(29,35,32),(30,36,33)], [(1,23),(2,24),(3,25),(4,26),(5,27),(6,19),(7,20),(8,21),(9,22),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36)], [(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)], [(1,12),(2,11),(3,10),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(19,34),(20,33),(21,32),(22,31),(23,30),(24,29),(25,28),(26,36),(27,35)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3H | 3I | 3J | 3K | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 9A | ··· | 9I | 9J | ··· | 9R | 18A | ··· | 18I |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 3 | 9 | 27 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 3 | 3 | 6 | 6 | 6 | 9 | 9 | 18 | 18 | 18 | 27 | 27 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | S3 | D6 | D6 | C3×S3 | D9 | C3×S3 | S3×C6 | D18 | S3×C6 | C3×D9 | C6×D9 | S32 | S3×D9 | C3×S32 | C3×S3×D9 |
| kernel | C3×S3×D9 | C32×D9 | S3×C3×C9 | C3×C9⋊S3 | S3×D9 | C3×D9 | S3×C9 | C9⋊S3 | C3×D9 | S3×C32 | C3×C9 | C33 | D9 | C3×S3 | C3×S3 | C9 | C32 | C32 | S3 | C3 | C32 | C3 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 2 | 3 | 2 | 6 | 6 | 1 | 3 | 2 | 6 |
Matrix representation of C3×S3×D9 ►in GL4(𝔽19) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 |
| 18 | 1 | 0 | 0 |
| 18 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 17 | 0 |
G:=sub<GL(4,GF(19))| [1,0,0,0,0,1,0,0,0,0,11,0,0,0,0,11],[18,18,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,18,0,0,0,0,18],[1,0,0,0,0,1,0,0,0,0,17,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,0,17,0,0,9,0] >;
C3×S3×D9 in GAP, Magma, Sage, TeX
C_3\times S_3\times D_9
% in TeX
G:=Group("C3xS3xD9"); // GroupNames label
G:=SmallGroup(324,114);
// by ID
G=gap.SmallGroup(324,114);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,1593,453,2164,3899]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^9=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations