direct product, metabelian, supersoluble, monomial, A-group
Aliases: C32×C3⋊Dic3, C34⋊6C4, C33⋊10C12, C33⋊9Dic3, C3⋊(C32×Dic3), C32⋊5(C3×C12), (C33×C6).2C2, C6.7(S3×C32), (C32×C6).21S3, (C32×C6).20C6, C32⋊6(C3×Dic3), C2.(C32×C3⋊S3), C6.16(C3×C3⋊S3), (C3×C6).22(C3×C6), (C3×C6).41(C3×S3), (C3×C6).27(C3⋊S3), SmallGroup(324,156)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C32×C3⋊Dic3 |
Generators and relations for C32×C3⋊Dic3
G = < a,b,c,d,e | a3=b3=c3=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >
Subgroups: 556 in 284 conjugacy classes, 78 normal (10 characteristic)
C1, C2, C3, C3, C4, C6, C6, C32, C32, C32, Dic3, C12, C3×C6, C3×C6, C3×C6, C33, C33, C3×Dic3, C3⋊Dic3, C3×C12, C32×C6, C32×C6, C34, C32×Dic3, C3×C3⋊Dic3, C33×C6, C32×C3⋊Dic3
Quotients: C1, C2, C3, C4, S3, C6, C32, Dic3, C12, C3×S3, C3⋊S3, C3×C6, C3×Dic3, C3⋊Dic3, C3×C12, S3×C32, C3×C3⋊S3, C32×Dic3, C3×C3⋊Dic3, C32×C3⋊S3, C32×C3⋊Dic3
►On 36 points
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 33 35)(32 34 36)
(1 22 27)(2 23 28)(3 24 29)(4 19 30)(5 20 25)(6 21 26)(7 18 33)(8 13 34)(9 14 35)(10 15 36)(11 16 31)(12 17 32)
(1 20 29)(2 21 30)(3 22 25)(4 23 26)(5 24 27)(6 19 28)(7 31 14)(8 32 15)(9 33 16)(10 34 17)(11 35 18)(12 36 13)
(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 7 4 10)(2 12 5 9)(3 11 6 8)(13 24 16 21)(14 23 17 20)(15 22 18 19)(25 35 28 32)(26 34 29 31)(27 33 30 36)
G:=sub<Sym(36)| (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(31,33,35)(32,34,36), (1,22,27)(2,23,28)(3,24,29)(4,19,30)(5,20,25)(6,21,26)(7,18,33)(8,13,34)(9,14,35)(10,15,36)(11,16,31)(12,17,32), (1,20,29)(2,21,30)(3,22,25)(4,23,26)(5,24,27)(6,19,28)(7,31,14)(8,32,15)(9,33,16)(10,34,17)(11,35,18)(12,36,13), (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), (1,7,4,10)(2,12,5,9)(3,11,6,8)(13,24,16,21)(14,23,17,20)(15,22,18,19)(25,35,28,32)(26,34,29,31)(27,33,30,36)>;
G:=Group( (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(31,33,35)(32,34,36), (1,22,27)(2,23,28)(3,24,29)(4,19,30)(5,20,25)(6,21,26)(7,18,33)(8,13,34)(9,14,35)(10,15,36)(11,16,31)(12,17,32), (1,20,29)(2,21,30)(3,22,25)(4,23,26)(5,24,27)(6,19,28)(7,31,14)(8,32,15)(9,33,16)(10,34,17)(11,35,18)(12,36,13), (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), (1,7,4,10)(2,12,5,9)(3,11,6,8)(13,24,16,21)(14,23,17,20)(15,22,18,19)(25,35,28,32)(26,34,29,31)(27,33,30,36) );
G=PermutationGroup([[(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,15,17),(14,16,18),(19,23,21),(20,24,22),(25,29,27),(26,30,28),(31,33,35),(32,34,36)], [(1,22,27),(2,23,28),(3,24,29),(4,19,30),(5,20,25),(6,21,26),(7,18,33),(8,13,34),(9,14,35),(10,15,36),(11,16,31),(12,17,32)], [(1,20,29),(2,21,30),(3,22,25),(4,23,26),(5,24,27),(6,19,28),(7,31,14),(8,32,15),(9,33,16),(10,34,17),(11,35,18),(12,36,13)], [(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)], [(1,7,4,10),(2,12,5,9),(3,11,6,8),(13,24,16,21),(14,23,17,20),(15,22,18,19),(25,35,28,32),(26,34,29,31),(27,33,30,36)]])
108 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | ··· | 3AR | 4A | 4B | 6A | ··· | 6H | 6I | ··· | 6AR | 12A | ··· | 12P |
| order | 1 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | 9 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | S3 | Dic3 | C3×S3 | C3×Dic3 |
| kernel | C32×C3⋊Dic3 | C33×C6 | C3×C3⋊Dic3 | C34 | C32×C6 | C33 | C32×C6 | C33 | C3×C6 | C32 |
| # reps | 1 | 1 | 8 | 2 | 8 | 16 | 4 | 4 | 32 | 32 |
Matrix representation of C32×C3⋊Dic3 ►in GL4(𝔽13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 3 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[9,0,0,0,0,9,0,0,0,0,3,0,0,0,0,3],[3,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,10,0,0,0,0,9,0,0,0,0,3],[0,12,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C32×C3⋊Dic3 in GAP, Magma, Sage, TeX
C_3^2\times C_3\rtimes {\rm Dic}_3 % in TeX
G:=Group("C3^2xC3:Dic3"); // GroupNames label
G:=SmallGroup(324,156);
// by ID
G=gap.SmallGroup(324,156);
# by ID
G:=PCGroup([6,-2,-3,-3,-2,-3,-3,108,2164,7781]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^6=1,e^2=d^3,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^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations