direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3:S3xC3xC6, C33:19D6, C34:8C22, C32:5C62, C6:(S3xC32), (C33xC6):2C2, (C32xC6):7S3, (C32xC6):8C6, C33:12(C2xC6), C32:10(S3xC6), C3:2(S3xC3xC6), (C3xC6):5(C3xS3), (C3xC6):4(C3xC6), SmallGroup(324,173)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C3:S3xC3xC6 |
Generators and relations for C3:S3xC3xC6
G = < a,b,c,d,e | a3=b6=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 820 in 356 conjugacy classes, 90 normal (10 characteristic)
C1, C2, C2, C3, C3, C22, S3, C6, C6, C32, C32, C32, D6, C2xC6, C3xS3, C3:S3, C3xC6, C3xC6, C3xC6, C33, C33, S3xC6, C2xC3:S3, C62, S3xC32, C3xC3:S3, C32xC6, C32xC6, C34, S3xC3xC6, C6xC3:S3, C32xC3:S3, C33xC6, C3:S3xC3xC6
Quotients: C1, C2, C3, C22, S3, C6, C32, D6, C2xC6, C3xS3, C3:S3, C3xC6, S3xC6, C2xC3:S3, C62, S3xC32, C3xC3:S3, S3xC3xC6, C6xC3:S3, C32xC3:S3, C3:S3xC3xC6
►On 36 points
(1 19 30)(2 20 25)(3 21 26)(4 22 27)(5 23 28)(6 24 29)(7 15 35)(8 16 36)(9 17 31)(10 18 32)(11 13 33)(12 14 34)
(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 23 26)(2 24 27)(3 19 28)(4 20 29)(5 21 30)(6 22 25)(7 31 13)(8 32 14)(9 33 15)(10 34 16)(11 35 17)(12 36 18)
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 35 33)(32 36 34)
(1 9)(2 10)(3 11)(4 12)(5 7)(6 8)(13 21)(14 22)(15 23)(16 24)(17 19)(18 20)(25 32)(26 33)(27 34)(28 35)(29 36)(30 31)
G:=sub<Sym(36)| (1,19,30)(2,20,25)(3,21,26)(4,22,27)(5,23,28)(6,24,29)(7,15,35)(8,16,36)(9,17,31)(10,18,32)(11,13,33)(12,14,34), (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), (1,23,26)(2,24,27)(3,19,28)(4,20,29)(5,21,30)(6,22,25)(7,31,13)(8,32,14)(9,33,15)(10,34,16)(11,35,17)(12,36,18), (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,35,33)(32,36,34), (1,9)(2,10)(3,11)(4,12)(5,7)(6,8)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20)(25,32)(26,33)(27,34)(28,35)(29,36)(30,31)>;
G:=Group( (1,19,30)(2,20,25)(3,21,26)(4,22,27)(5,23,28)(6,24,29)(7,15,35)(8,16,36)(9,17,31)(10,18,32)(11,13,33)(12,14,34), (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), (1,23,26)(2,24,27)(3,19,28)(4,20,29)(5,21,30)(6,22,25)(7,31,13)(8,32,14)(9,33,15)(10,34,16)(11,35,17)(12,36,18), (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,35,33)(32,36,34), (1,9)(2,10)(3,11)(4,12)(5,7)(6,8)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20)(25,32)(26,33)(27,34)(28,35)(29,36)(30,31) );
G=PermutationGroup([[(1,19,30),(2,20,25),(3,21,26),(4,22,27),(5,23,28),(6,24,29),(7,15,35),(8,16,36),(9,17,31),(10,18,32),(11,13,33),(12,14,34)], [(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)], [(1,23,26),(2,24,27),(3,19,28),(4,20,29),(5,21,30),(6,22,25),(7,31,13),(8,32,14),(9,33,15),(10,34,16),(11,35,17),(12,36,18)], [(1,3,5),(2,4,6),(7,11,9),(8,12,10),(13,17,15),(14,18,16),(19,21,23),(20,22,24),(25,27,29),(26,28,30),(31,35,33),(32,36,34)], [(1,9),(2,10),(3,11),(4,12),(5,7),(6,8),(13,21),(14,22),(15,23),(16,24),(17,19),(18,20),(25,32),(26,33),(27,34),(28,35),(29,36),(30,31)]])
108 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 3I | ··· | 3AR | 6A | ··· | 6H | 6I | ··· | 6AR | 6AS | ··· | 6BH |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 |
| size | 1 | 1 | 9 | 9 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | D6 | C3xS3 | S3xC6 |
| kernel | C3:S3xC3xC6 | C32xC3:S3 | C33xC6 | C6xC3:S3 | C3xC3:S3 | C32xC6 | C32xC6 | C33 | C3xC6 | C32 |
| # reps | 1 | 2 | 1 | 8 | 16 | 8 | 4 | 4 | 32 | 32 |
Matrix representation of C3:S3xC3xC6 ►in GL4(F7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 6 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 4 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,2,0,0,0,0,2],[6,0,0,0,0,6,0,0,0,0,3,0,0,0,0,3],[4,0,0,0,0,2,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,2,0,0,0,0,4],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C3:S3xC3xC6 in GAP, Magma, Sage, TeX
C_3\rtimes S_3\times C_3\times C_6
% in TeX
G:=Group("C3:S3xC3xC6"); // GroupNames label
G:=SmallGroup(324,173);
// by ID
G=gap.SmallGroup(324,173);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,2164,7781]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^6=c^3=d^3=e^2=1,a*b=b*a,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=c^-1,e*d*e=d^-1>;
// generators/relations