direct product, metabelian, supersoluble, monomial, A-group
Aliases: S3×C3×C12, C6.2C62, D6.(C3×C6), C12⋊2(C3×C6), (C3×C12)⋊7C6, C3⋊1(C6×C12), (S3×C6).6C6, C6.36(S3×C6), C33⋊10(C2×C4), (C3×C6).68D6, C32⋊7(C2×C12), (C32×C12)⋊4C2, Dic3⋊2(C3×C6), (C3×Dic3)⋊5C6, (C32×Dic3)⋊8C2, (C32×C6).17C22, C2.1(S3×C3×C6), (S3×C3×C6).3C2, (C3×C6).25(C2×C6), (C3×Dic3)○(C3×C12), (C3×C12)○(C32×Dic3), SmallGroup(216,136)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C3×C12 |
Generators and relations for S3×C3×C12
G = < a,b,c,d | a3=b12=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 204 in 120 conjugacy classes, 66 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C2×C4, C32, C32, C32, Dic3, C12, C12, C12, D6, C2×C6, C3×S3, C3×C6, C3×C6, C3×C6, C4×S3, C2×C12, C33, C3×Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C62, S3×C32, C32×C6, S3×C12, C6×C12, C32×Dic3, C32×C12, S3×C3×C6, S3×C3×C12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C32, C12, D6, C2×C6, C3×S3, C3×C6, C4×S3, C2×C12, C3×C12, S3×C6, C62, S3×C32, S3×C12, C6×C12, S3×C3×C6, S3×C3×C12
►On 72 points
(1 40 29)(2 41 30)(3 42 31)(4 43 32)(5 44 33)(6 45 34)(7 46 35)(8 47 36)(9 48 25)(10 37 26)(11 38 27)(12 39 28)(13 56 72)(14 57 61)(15 58 62)(16 59 63)(17 60 64)(18 49 65)(19 50 66)(20 51 67)(21 52 68)(22 53 69)(23 54 70)(24 55 71)
(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 48 33)(2 37 34)(3 38 35)(4 39 36)(5 40 25)(6 41 26)(7 42 27)(8 43 28)(9 44 29)(10 45 30)(11 46 31)(12 47 32)(13 64 52)(14 65 53)(15 66 54)(16 67 55)(17 68 56)(18 69 57)(19 70 58)(20 71 59)(21 72 60)(22 61 49)(23 62 50)(24 63 51)
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 45)(14 46)(15 47)(16 48)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 59)(26 60)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 55)(34 56)(35 57)(36 58)
G:=sub<Sym(72)| (1,40,29)(2,41,30)(3,42,31)(4,43,32)(5,44,33)(6,45,34)(7,46,35)(8,47,36)(9,48,25)(10,37,26)(11,38,27)(12,39,28)(13,56,72)(14,57,61)(15,58,62)(16,59,63)(17,60,64)(18,49,65)(19,50,66)(20,51,67)(21,52,68)(22,53,69)(23,54,70)(24,55,71), (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), (1,48,33)(2,37,34)(3,38,35)(4,39,36)(5,40,25)(6,41,26)(7,42,27)(8,43,28)(9,44,29)(10,45,30)(11,46,31)(12,47,32)(13,64,52)(14,65,53)(15,66,54)(16,67,55)(17,68,56)(18,69,57)(19,70,58)(20,71,59)(21,72,60)(22,61,49)(23,62,50)(24,63,51), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,45)(14,46)(15,47)(16,48)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,59)(26,60)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,57)(36,58)>;
G:=Group( (1,40,29)(2,41,30)(3,42,31)(4,43,32)(5,44,33)(6,45,34)(7,46,35)(8,47,36)(9,48,25)(10,37,26)(11,38,27)(12,39,28)(13,56,72)(14,57,61)(15,58,62)(16,59,63)(17,60,64)(18,49,65)(19,50,66)(20,51,67)(21,52,68)(22,53,69)(23,54,70)(24,55,71), (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), (1,48,33)(2,37,34)(3,38,35)(4,39,36)(5,40,25)(6,41,26)(7,42,27)(8,43,28)(9,44,29)(10,45,30)(11,46,31)(12,47,32)(13,64,52)(14,65,53)(15,66,54)(16,67,55)(17,68,56)(18,69,57)(19,70,58)(20,71,59)(21,72,60)(22,61,49)(23,62,50)(24,63,51), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,45)(14,46)(15,47)(16,48)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,59)(26,60)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,57)(36,58) );
G=PermutationGroup([[(1,40,29),(2,41,30),(3,42,31),(4,43,32),(5,44,33),(6,45,34),(7,46,35),(8,47,36),(9,48,25),(10,37,26),(11,38,27),(12,39,28),(13,56,72),(14,57,61),(15,58,62),(16,59,63),(17,60,64),(18,49,65),(19,50,66),(20,51,67),(21,52,68),(22,53,69),(23,54,70),(24,55,71)], [(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)], [(1,48,33),(2,37,34),(3,38,35),(4,39,36),(5,40,25),(6,41,26),(7,42,27),(8,43,28),(9,44,29),(10,45,30),(11,46,31),(12,47,32),(13,64,52),(14,65,53),(15,66,54),(16,67,55),(17,68,56),(18,69,57),(19,70,58),(20,71,59),(21,72,60),(22,61,49),(23,62,50),(24,63,51)], [(1,67),(2,68),(3,69),(4,70),(5,71),(6,72),(7,61),(8,62),(9,63),(10,64),(11,65),(12,66),(13,45),(14,46),(15,47),(16,48),(17,37),(18,38),(19,39),(20,40),(21,41),(22,42),(23,43),(24,44),(25,59),(26,60),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(33,55),(34,56),(35,57),(36,58)]])
S3×C3×C12 is a maximal subgroup of
C33⋊7M4(2) C12.73S32 C12.57S32 C12.58S32
108 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 3I | ··· | 3Q | 4A | 4B | 4C | 4D | 6A | ··· | 6H | 6I | ··· | 6Q | 6R | ··· | 6AG | 12A | ··· | 12P | 12Q | ··· | 12AH | 12AI | ··· | 12AX |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 3 | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 3 | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | S3 | D6 | C3×S3 | C4×S3 | S3×C6 | S3×C12 |
| kernel | S3×C3×C12 | C32×Dic3 | C32×C12 | S3×C3×C6 | S3×C12 | S3×C32 | C3×Dic3 | C3×C12 | S3×C6 | C3×S3 | C3×C12 | C3×C6 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 8 | 4 | 8 | 8 | 8 | 32 | 1 | 1 | 8 | 2 | 8 | 16 |
Matrix representation of S3×C3×C12 ►in GL3(𝔽13) generated by
| 1 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 3 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 1 | 0 | 0 |
| 0 | 3 | 3 |
| 0 | 0 | 9 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 2 | 12 |
G:=sub<GL(3,GF(13))| [1,0,0,0,3,0,0,0,3],[3,0,0,0,8,0,0,0,8],[1,0,0,0,3,0,0,3,9],[1,0,0,0,1,2,0,0,12] >;
S3×C3×C12 in GAP, Magma, Sage, TeX
S_3\times C_3\times C_{12} % in TeX
G:=Group("S3xC3xC12"); // GroupNames label
G:=SmallGroup(216,136);
// by ID
G=gap.SmallGroup(216,136);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-2,-3,223,5189]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^12=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations