direct product, metabelian, supersoluble, monomial
Aliases: C3xC12:S3, C33:11D4, C32:7D12, C12:1(C3xS3), (C3xC12):5C6, (C3xC12):6S3, C3:1(C3xD12), C12:3(C3:S3), C6.26(S3xC6), (C3xC6).59D6, C32:8(C3xD4), (C32xC12):3C2, (C32xC6).23C22, C4:(C3xC3:S3), (C6xC3:S3):5C2, (C2xC3:S3):5C6, C2.4(C6xC3:S3), C6.24(C2xC3:S3), (C3xC6).31(C2xC6), SmallGroup(216,142)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3xC12:S3
G = < a,b,c,d | a3=b12=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 392 in 120 conjugacy classes, 42 normal (14 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, D4, C32, C32, C32, C12, C12, C12, D6, C2xC6, C3xS3, C3:S3, C3xC6, C3xC6, C3xC6, D12, C3xD4, C33, C3xC12, C3xC12, C3xC12, S3xC6, C2xC3:S3, C3xC3:S3, C32xC6, C3xD12, C12:S3, C32xC12, C6xC3:S3, C3xC12:S3
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2xC6, C3xS3, C3:S3, D12, C3xD4, S3xC6, C2xC3:S3, C3xC3:S3, C3xD12, C12:S3, C6xC3:S3, C3xC12:S3
►On 72 points
(1 50 67)(2 51 68)(3 52 69)(4 53 70)(5 54 71)(6 55 72)(7 56 61)(8 57 62)(9 58 63)(10 59 64)(11 60 65)(12 49 66)(13 29 48)(14 30 37)(15 31 38)(16 32 39)(17 33 40)(18 34 41)(19 35 42)(20 36 43)(21 25 44)(22 26 45)(23 27 46)(24 28 47)
(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 54 63)(2 55 64)(3 56 65)(4 57 66)(5 58 67)(6 59 68)(7 60 69)(8 49 70)(9 50 71)(10 51 72)(11 52 61)(12 53 62)(13 40 25)(14 41 26)(15 42 27)(16 43 28)(17 44 29)(18 45 30)(19 46 31)(20 47 32)(21 48 33)(22 37 34)(23 38 35)(24 39 36)
(1 41)(2 40)(3 39)(4 38)(5 37)(6 48)(7 47)(8 46)(9 45)(10 44)(11 43)(12 42)(13 55)(14 54)(15 53)(16 52)(17 51)(18 50)(19 49)(20 60)(21 59)(22 58)(23 57)(24 56)(25 64)(26 63)(27 62)(28 61)(29 72)(30 71)(31 70)(32 69)(33 68)(34 67)(35 66)(36 65)
G:=sub<Sym(72)| (1,50,67)(2,51,68)(3,52,69)(4,53,70)(5,54,71)(6,55,72)(7,56,61)(8,57,62)(9,58,63)(10,59,64)(11,60,65)(12,49,66)(13,29,48)(14,30,37)(15,31,38)(16,32,39)(17,33,40)(18,34,41)(19,35,42)(20,36,43)(21,25,44)(22,26,45)(23,27,46)(24,28,47), (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), (1,54,63)(2,55,64)(3,56,65)(4,57,66)(5,58,67)(6,59,68)(7,60,69)(8,49,70)(9,50,71)(10,51,72)(11,52,61)(12,53,62)(13,40,25)(14,41,26)(15,42,27)(16,43,28)(17,44,29)(18,45,30)(19,46,31)(20,47,32)(21,48,33)(22,37,34)(23,38,35)(24,39,36), (1,41)(2,40)(3,39)(4,38)(5,37)(6,48)(7,47)(8,46)(9,45)(10,44)(11,43)(12,42)(13,55)(14,54)(15,53)(16,52)(17,51)(18,50)(19,49)(20,60)(21,59)(22,58)(23,57)(24,56)(25,64)(26,63)(27,62)(28,61)(29,72)(30,71)(31,70)(32,69)(33,68)(34,67)(35,66)(36,65)>;
G:=Group( (1,50,67)(2,51,68)(3,52,69)(4,53,70)(5,54,71)(6,55,72)(7,56,61)(8,57,62)(9,58,63)(10,59,64)(11,60,65)(12,49,66)(13,29,48)(14,30,37)(15,31,38)(16,32,39)(17,33,40)(18,34,41)(19,35,42)(20,36,43)(21,25,44)(22,26,45)(23,27,46)(24,28,47), (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), (1,54,63)(2,55,64)(3,56,65)(4,57,66)(5,58,67)(6,59,68)(7,60,69)(8,49,70)(9,50,71)(10,51,72)(11,52,61)(12,53,62)(13,40,25)(14,41,26)(15,42,27)(16,43,28)(17,44,29)(18,45,30)(19,46,31)(20,47,32)(21,48,33)(22,37,34)(23,38,35)(24,39,36), (1,41)(2,40)(3,39)(4,38)(5,37)(6,48)(7,47)(8,46)(9,45)(10,44)(11,43)(12,42)(13,55)(14,54)(15,53)(16,52)(17,51)(18,50)(19,49)(20,60)(21,59)(22,58)(23,57)(24,56)(25,64)(26,63)(27,62)(28,61)(29,72)(30,71)(31,70)(32,69)(33,68)(34,67)(35,66)(36,65) );
G=PermutationGroup([[(1,50,67),(2,51,68),(3,52,69),(4,53,70),(5,54,71),(6,55,72),(7,56,61),(8,57,62),(9,58,63),(10,59,64),(11,60,65),(12,49,66),(13,29,48),(14,30,37),(15,31,38),(16,32,39),(17,33,40),(18,34,41),(19,35,42),(20,36,43),(21,25,44),(22,26,45),(23,27,46),(24,28,47)], [(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)], [(1,54,63),(2,55,64),(3,56,65),(4,57,66),(5,58,67),(6,59,68),(7,60,69),(8,49,70),(9,50,71),(10,51,72),(11,52,61),(12,53,62),(13,40,25),(14,41,26),(15,42,27),(16,43,28),(17,44,29),(18,45,30),(19,46,31),(20,47,32),(21,48,33),(22,37,34),(23,38,35),(24,39,36)], [(1,41),(2,40),(3,39),(4,38),(5,37),(6,48),(7,47),(8,46),(9,45),(10,44),(11,43),(12,42),(13,55),(14,54),(15,53),(16,52),(17,51),(18,50),(19,49),(20,60),(21,59),(22,58),(23,57),(24,56),(25,64),(26,63),(27,62),(28,61),(29,72),(30,71),(31,70),(32,69),(33,68),(34,67),(35,66),(36,65)]])
C3xC12:S3 is a maximal subgroup of
C33:6D8 C33:8D8 C33:13SD16 C33:16SD16 C33:9D8 C33:18SD16 C3xS3xD12 C12.39S32 C12.57S32 C12:S32 C12:S3:12S3 C12:3S32 C3xD4xC3:S3
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3N | 4 | 6A | 6B | 6C | ··· | 6N | 6O | 6P | 6Q | 6R | 12A | ··· | 12Z |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 18 | 18 | 1 | 1 | 2 | ··· | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 18 | 18 | 18 | 18 | 2 | ··· | 2 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | D4 | D6 | C3xS3 | D12 | C3xD4 | S3xC6 | C3xD12 |
| kernel | C3xC12:S3 | C32xC12 | C6xC3:S3 | C12:S3 | C3xC12 | C2xC3:S3 | C3xC12 | C33 | C3xC6 | C12 | C32 | C32 | C6 | C3 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 4 | 8 | 8 | 2 | 8 | 16 |
Matrix representation of C3xC12:S3 ►in GL4(F13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 10 | 10 | 0 | 0 |
| 3 | 7 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 5 |
| 12 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 3 |
| 10 | 10 | 0 | 0 |
| 7 | 3 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[10,3,0,0,10,7,0,0,0,0,8,0,0,0,0,5],[12,12,0,0,1,0,0,0,0,0,9,0,0,0,0,3],[10,7,0,0,10,3,0,0,0,0,0,1,0,0,1,0] >;
C3xC12:S3 in GAP, Magma, Sage, TeX
C_3\times C_{12}\rtimes S_3 % in TeX
G:=Group("C3xC12:S3"); // GroupNames label
G:=SmallGroup(216,142);
// by ID
G=gap.SmallGroup(216,142);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,79,1444,5189]);
// 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,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations