direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3×C3⋊Dic3, C33⋊4C4, C32⋊5C12, C32⋊4Dic3, C3⋊(C3×Dic3), C6.7(C3×S3), (C3×C6).9S3, (C3×C6).8C6, C6.6(C3⋊S3), (C32×C6).2C2, C2.(C3×C3⋊S3), SmallGroup(108,33)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C3×C3⋊Dic3 |
Generators and relations for C3×C3⋊Dic3
G = < a,b,c,d | a3=b3=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 100 in 52 conjugacy classes, 26 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⋊Dic3, C32×C6, C3×C3⋊Dic3
Quotients: C1, C2, C3, C4, S3, C6, Dic3, C12, C3×S3, C3⋊S3, C3×Dic3, C3⋊Dic3, C3×C3⋊S3, C3×C3⋊Dic3
►On 36 points
(1 25 24)(2 26 19)(3 27 20)(4 28 21)(5 29 22)(6 30 23)(7 35 16)(8 36 17)(9 31 18)(10 32 13)(11 33 14)(12 34 15)
(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)| (1,25,24)(2,26,19)(3,27,20)(4,28,21)(5,29,22)(6,30,23)(7,35,16)(8,36,17)(9,31,18)(10,32,13)(11,33,14)(12,34,15), (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( (1,25,24)(2,26,19)(3,27,20)(4,28,21)(5,29,22)(6,30,23)(7,35,16)(8,36,17)(9,31,18)(10,32,13)(11,33,14)(12,34,15), (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([[(1,25,24),(2,26,19),(3,27,20),(4,28,21),(5,29,22),(6,30,23),(7,35,16),(8,36,17),(9,31,18),(10,32,13),(11,33,14),(12,34,15)], [(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)]])
C3×C3⋊Dic3 is a maximal subgroup of
C33⋊4C8 C3×S3×Dic3 C33⋊8(C2×C4) C33⋊7D4 C33⋊4Q8 C33⋊9(C2×C4) C33⋊9D4 C33⋊5Q8 C12×C3⋊S3 C32⋊C36 C32⋊Dic9 C32⋊2Dic9 C33⋊Dic3 C33⋊4C12 C33.Dic3 He3⋊6Dic3 C3⋊Dic3.2A4
C3×C3⋊Dic3 is a maximal quotient of
C33⋊4C12 C33.Dic3 He3.4Dic3 He3.5C12
36 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3N | 4A | 4B | 6A | 6B | 6C | ··· | 6N | 12A | 12B | 12C | 12D |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 9 | 9 | 1 | 1 | 2 | ··· | 2 | 9 | 9 | 9 | 9 |
36 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 | C3×C3⋊Dic3 | C32×C6 | C3⋊Dic3 | C33 | C3×C6 | C32 | C3×C6 | C32 | C6 | C3 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 |
Matrix representation of C3×C3⋊Dic3 ►in GL4(𝔽13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 3 |
| 10 | 0 | 0 | 0 |
| 2 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 8 | 2 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,3],[10,2,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[8,0,0,0,2,5,0,0,0,0,0,1,0,0,1,0] >;
C3×C3⋊Dic3 in GAP, Magma, Sage, TeX
C_3\times C_3\rtimes {\rm Dic}_3 % in TeX
G:=Group("C3xC3:Dic3"); // GroupNames label
G:=SmallGroup(108,33);
// by ID
G=gap.SmallGroup(108,33);
# by ID
G:=PCGroup([5,-2,-3,-2,-3,-3,30,483,1804]);
// 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,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations