direct product, metabelian, supersoluble, monomial, A-group
Aliases: Dic3×C3×C6, C6.4C62, C62.17S3, C62.13C6, C6⋊(C3×C12), C3⋊2(C6×C12), (C3×C6)⋊4C12, C6.38(S3×C6), (C32×C6)⋊3C4, C33⋊13(C2×C4), (C3×C6).70D6, C32⋊9(C2×C12), (C3×C62).1C2, C22.(S3×C32), (C32×C6).19C22, C2.2(S3×C3×C6), (C2×C6).9(C3×C6), (C2×C6).22(C3×S3), (C3×C6).27(C2×C6), SmallGroup(216,138)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — Dic3×C3×C6 |
Generators and relations for Dic3×C3×C6
G = < a,b,c,d | a3=b6=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 212 in 136 conjugacy classes, 78 normal (14 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C6, C6, C6, C2×C4, C32, C32, C32, Dic3, C12, C2×C6, C2×C6, C2×C6, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C33, C3×Dic3, C3×C12, C62, C62, C62, C32×C6, C32×C6, C6×Dic3, C6×C12, C32×Dic3, C3×C62, Dic3×C3×C6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C32, Dic3, C12, D6, C2×C6, C3×S3, C3×C6, C2×Dic3, C2×C12, C3×Dic3, C3×C12, S3×C6, C62, S3×C32, C6×Dic3, C6×C12, C32×Dic3, S3×C3×C6, Dic3×C3×C6
►On 72 points
(1 63 43)(2 64 44)(3 65 45)(4 66 46)(5 61 47)(6 62 48)(7 17 33)(8 18 34)(9 13 35)(10 14 36)(11 15 31)(12 16 32)(19 70 29)(20 71 30)(21 72 25)(22 67 26)(23 68 27)(24 69 28)(37 51 60)(38 52 55)(39 53 56)(40 54 57)(41 49 58)(42 50 59)
(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)(37 38 39 40 41 42)(43 44 45 46 47 48)(49 50 51 52 53 54)(55 56 57 58 59 60)(61 62 63 64 65 66)(67 68 69 70 71 72)
(1 37 65 53 47 58)(2 38 66 54 48 59)(3 39 61 49 43 60)(4 40 62 50 44 55)(5 41 63 51 45 56)(6 42 64 52 46 57)(7 72 31 19 13 27)(8 67 32 20 14 28)(9 68 33 21 15 29)(10 69 34 22 16 30)(11 70 35 23 17 25)(12 71 36 24 18 26)
(1 24 53 12)(2 19 54 7)(3 20 49 8)(4 21 50 9)(5 22 51 10)(6 23 52 11)(13 66 72 59)(14 61 67 60)(15 62 68 55)(16 63 69 56)(17 64 70 57)(18 65 71 58)(25 42 35 46)(26 37 36 47)(27 38 31 48)(28 39 32 43)(29 40 33 44)(30 41 34 45)
G:=sub<Sym(72)| (1,63,43)(2,64,44)(3,65,45)(4,66,46)(5,61,47)(6,62,48)(7,17,33)(8,18,34)(9,13,35)(10,14,36)(11,15,31)(12,16,32)(19,70,29)(20,71,30)(21,72,25)(22,67,26)(23,68,27)(24,69,28)(37,51,60)(38,52,55)(39,53,56)(40,54,57)(41,49,58)(42,50,59), (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)(37,38,39,40,41,42)(43,44,45,46,47,48)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72), (1,37,65,53,47,58)(2,38,66,54,48,59)(3,39,61,49,43,60)(4,40,62,50,44,55)(5,41,63,51,45,56)(6,42,64,52,46,57)(7,72,31,19,13,27)(8,67,32,20,14,28)(9,68,33,21,15,29)(10,69,34,22,16,30)(11,70,35,23,17,25)(12,71,36,24,18,26), (1,24,53,12)(2,19,54,7)(3,20,49,8)(4,21,50,9)(5,22,51,10)(6,23,52,11)(13,66,72,59)(14,61,67,60)(15,62,68,55)(16,63,69,56)(17,64,70,57)(18,65,71,58)(25,42,35,46)(26,37,36,47)(27,38,31,48)(28,39,32,43)(29,40,33,44)(30,41,34,45)>;
G:=Group( (1,63,43)(2,64,44)(3,65,45)(4,66,46)(5,61,47)(6,62,48)(7,17,33)(8,18,34)(9,13,35)(10,14,36)(11,15,31)(12,16,32)(19,70,29)(20,71,30)(21,72,25)(22,67,26)(23,68,27)(24,69,28)(37,51,60)(38,52,55)(39,53,56)(40,54,57)(41,49,58)(42,50,59), (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)(37,38,39,40,41,42)(43,44,45,46,47,48)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72), (1,37,65,53,47,58)(2,38,66,54,48,59)(3,39,61,49,43,60)(4,40,62,50,44,55)(5,41,63,51,45,56)(6,42,64,52,46,57)(7,72,31,19,13,27)(8,67,32,20,14,28)(9,68,33,21,15,29)(10,69,34,22,16,30)(11,70,35,23,17,25)(12,71,36,24,18,26), (1,24,53,12)(2,19,54,7)(3,20,49,8)(4,21,50,9)(5,22,51,10)(6,23,52,11)(13,66,72,59)(14,61,67,60)(15,62,68,55)(16,63,69,56)(17,64,70,57)(18,65,71,58)(25,42,35,46)(26,37,36,47)(27,38,31,48)(28,39,32,43)(29,40,33,44)(30,41,34,45) );
G=PermutationGroup([[(1,63,43),(2,64,44),(3,65,45),(4,66,46),(5,61,47),(6,62,48),(7,17,33),(8,18,34),(9,13,35),(10,14,36),(11,15,31),(12,16,32),(19,70,29),(20,71,30),(21,72,25),(22,67,26),(23,68,27),(24,69,28),(37,51,60),(38,52,55),(39,53,56),(40,54,57),(41,49,58),(42,50,59)], [(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),(37,38,39,40,41,42),(43,44,45,46,47,48),(49,50,51,52,53,54),(55,56,57,58,59,60),(61,62,63,64,65,66),(67,68,69,70,71,72)], [(1,37,65,53,47,58),(2,38,66,54,48,59),(3,39,61,49,43,60),(4,40,62,50,44,55),(5,41,63,51,45,56),(6,42,64,52,46,57),(7,72,31,19,13,27),(8,67,32,20,14,28),(9,68,33,21,15,29),(10,69,34,22,16,30),(11,70,35,23,17,25),(12,71,36,24,18,26)], [(1,24,53,12),(2,19,54,7),(3,20,49,8),(4,21,50,9),(5,22,51,10),(6,23,52,11),(13,66,72,59),(14,61,67,60),(15,62,68,55),(16,63,69,56),(17,64,70,57),(18,65,71,58),(25,42,35,46),(26,37,36,47),(27,38,31,48),(28,39,32,43),(29,40,33,44),(30,41,34,45)]])
Dic3×C3×C6 is a maximal subgroup of
C62.78D6 C62.79D6 C62.80D6 C62.81D6 C62.82D6 C62.93D6 S3×C6×C12
108 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 3I | ··· | 3Q | 4A | 4B | 4C | 4D | 6A | ··· | 6X | 6Y | ··· | 6AY | 12A | ··· | 12AF |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | S3 | Dic3 | D6 | C3×S3 | C3×Dic3 | S3×C6 |
| kernel | Dic3×C3×C6 | C32×Dic3 | C3×C62 | C6×Dic3 | C32×C6 | C3×Dic3 | C62 | C3×C6 | C62 | C3×C6 | C3×C6 | C2×C6 | C6 | C6 |
| # reps | 1 | 2 | 1 | 8 | 4 | 16 | 8 | 32 | 1 | 2 | 1 | 8 | 16 | 8 |
Matrix representation of Dic3×C3×C6 ►in GL3(𝔽13) generated by
| 9 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 9 |
| 12 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 10 |
| 12 | 0 | 0 |
| 0 | 9 | 3 |
| 0 | 0 | 3 |
| 5 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 2 | 1 |
G:=sub<GL(3,GF(13))| [9,0,0,0,9,0,0,0,9],[12,0,0,0,10,0,0,0,10],[12,0,0,0,9,0,0,3,3],[5,0,0,0,12,2,0,0,1] >;
Dic3×C3×C6 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_3\times C_6 % in TeX
G:=Group("Dic3xC3xC6"); // GroupNames label
G:=SmallGroup(216,138);
// by ID
G=gap.SmallGroup(216,138);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-2,-3,216,5189]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^6=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