direct product, metabelian, supersoluble, monomial, A-group
Aliases: C32×Dic3, C33⋊3C4, C32⋊4C12, C3⋊(C3×C12), C6.(C3×C6), (C3×C6).7C6, C2.(S3×C32), C6.10(C3×S3), (C3×C6).11S3, (C32×C6).1C2, SmallGroup(108,32)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C32×Dic3 |
Generators and relations for C32×Dic3
G = < a,b,c,d | a3=b3=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 80 in 52 conjugacy classes, 30 normal (10 characteristic)
C1, C2, C3, C3, C3, C4, C6, C6, C6, C32, C32, C32, Dic3, C12, C3×C6, C3×C6, C3×C6, C33, C3×Dic3, C3×C12, C32×C6, C32×Dic3
Quotients: C1, C2, C3, C4, S3, C6, C32, Dic3, C12, C3×S3, C3×C6, C3×Dic3, C3×C12, S3×C32, C32×Dic3
►On 36 points
(1 15 7)(2 16 8)(3 17 9)(4 18 10)(5 13 11)(6 14 12)(19 33 29)(20 34 30)(21 35 25)(22 36 26)(23 31 27)(24 32 28)
(1 3 5)(2 4 6)(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 35 33)(32 36 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 22 4 19)(2 21 5 24)(3 20 6 23)(7 26 10 29)(8 25 11 28)(9 30 12 27)(13 32 16 35)(14 31 17 34)(15 36 18 33)
G:=sub<Sym(36)| (1,15,7)(2,16,8)(3,17,9)(4,18,10)(5,13,11)(6,14,12)(19,33,29)(20,34,30)(21,35,25)(22,36,26)(23,31,27)(24,32,28), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(31,35,33)(32,36,34), (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), (1,22,4,19)(2,21,5,24)(3,20,6,23)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,32,16,35)(14,31,17,34)(15,36,18,33)>;
G:=Group( (1,15,7)(2,16,8)(3,17,9)(4,18,10)(5,13,11)(6,14,12)(19,33,29)(20,34,30)(21,35,25)(22,36,26)(23,31,27)(24,32,28), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(31,35,33)(32,36,34), (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), (1,22,4,19)(2,21,5,24)(3,20,6,23)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,32,16,35)(14,31,17,34)(15,36,18,33) );
G=PermutationGroup([[(1,15,7),(2,16,8),(3,17,9),(4,18,10),(5,13,11),(6,14,12),(19,33,29),(20,34,30),(21,35,25),(22,36,26),(23,31,27),(24,32,28)], [(1,3,5),(2,4,6),(7,9,11),(8,10,12),(13,15,17),(14,16,18),(19,23,21),(20,24,22),(25,29,27),(26,30,28),(31,35,33),(32,36,34)], [(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)], [(1,22,4,19),(2,21,5,24),(3,20,6,23),(7,26,10,29),(8,25,11,28),(9,30,12,27),(13,32,16,35),(14,31,17,34),(15,36,18,33)]])
C32×Dic3 is a maximal subgroup of
C33⋊8(C2×C4) C33⋊8D4 C33⋊4Q8 S3×C3×C12 He3⋊C12
54 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | ··· | 3Q | 4A | 4B | 6A | ··· | 6H | 6I | ··· | 6Q | 12A | ··· | 12P |
| order | 1 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
54 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×Dic3 | C32×C6 | C3×Dic3 | C33 | C3×C6 | C32 | C3×C6 | C32 | C6 | C3 |
| # reps | 1 | 1 | 8 | 2 | 8 | 16 | 1 | 1 | 8 | 8 |
Matrix representation of C32×Dic3 ►in GL3(𝔽13) generated by
| 3 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 3 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 1 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 10 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 12 | 0 |
G:=sub<GL(3,GF(13))| [3,0,0,0,1,0,0,0,1],[3,0,0,0,3,0,0,0,3],[1,0,0,0,4,0,0,0,10],[1,0,0,0,0,12,0,1,0] >;
C32×Dic3 in GAP, Magma, Sage, TeX
C_3^2\times {\rm Dic}_3 % in TeX
G:=Group("C3^2xDic3"); // GroupNames label
G:=SmallGroup(108,32);
// by ID
G=gap.SmallGroup(108,32);
# by ID
G:=PCGroup([5,-2,-3,-3,-2,-3,90,1804]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations