direct product, metabelian, supersoluble, monomial, A-group
Aliases: S3×C4×C12, C122⋊13C2, C62.162C23, (C4×C12)⋊14C6, C12⋊5(C2×C12), D6.6(C2×C12), C32⋊5(C2×C42), (C4×Dic3)⋊17C6, Dic3⋊5(C2×C12), (C2×C12).458D6, C6.2(C22×C12), (Dic3×C12)⋊35C2, (C6×C12).344C22, (C6×Dic3).165C22, C3⋊1(C2×C4×C12), C2.1(S3×C2×C12), (S3×C2×C4).11C6, C6.101(S3×C2×C4), (C3×C12)⋊19(C2×C4), C22.9(S3×C2×C6), (S3×C2×C12).24C2, (C2×C4).96(S3×C6), (S3×C6).25(C2×C4), (C2×C12).126(C2×C6), (C3×Dic3)⋊18(C2×C4), (S3×C2×C6).113C22, (C2×C6).17(C22×C6), (C3×C6).73(C22×C4), (C22×S3).33(C2×C6), (C2×C6).295(C22×S3), (C2×Dic3).46(C2×C6), SmallGroup(288,642)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C4×C12 |
Generators and relations for S3×C4×C12
G = < a,b,c,d | a4=b12=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 402 in 231 conjugacy classes, 138 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C42, C22×C4, C3×S3, C3×C6, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, C2×C42, C3×Dic3, C3×C12, S3×C6, C62, C4×Dic3, C4×C12, C4×C12, S3×C2×C4, C22×C12, S3×C12, C6×Dic3, C6×C12, S3×C2×C6, S3×C42, C2×C4×C12, Dic3×C12, C122, S3×C2×C12, S3×C4×C12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, C12, D6, C2×C6, C42, C22×C4, C3×S3, C4×S3, C2×C12, C22×S3, C22×C6, C2×C42, S3×C6, C4×C12, S3×C2×C4, C22×C12, S3×C12, S3×C2×C6, S3×C42, C2×C4×C12, S3×C2×C12, S3×C4×C12
►On 96 points
(1 41 35 56)(2 42 36 57)(3 43 25 58)(4 44 26 59)(5 45 27 60)(6 46 28 49)(7 47 29 50)(8 48 30 51)(9 37 31 52)(10 38 32 53)(11 39 33 54)(12 40 34 55)(13 63 96 78)(14 64 85 79)(15 65 86 80)(16 66 87 81)(17 67 88 82)(18 68 89 83)(19 69 90 84)(20 70 91 73)(21 71 92 74)(22 72 93 75)(23 61 94 76)(24 62 95 77)
(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)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 41 45)(38 42 46)(39 43 47)(40 44 48)(49 53 57)(50 54 58)(51 55 59)(52 56 60)(61 69 65)(62 70 66)(63 71 67)(64 72 68)(73 81 77)(74 82 78)(75 83 79)(76 84 80)(85 93 89)(86 94 90)(87 95 91)(88 96 92)
(1 68)(2 69)(3 70)(4 71)(5 72)(6 61)(7 62)(8 63)(9 64)(10 65)(11 66)(12 67)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 49)(24 50)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)
G:=sub<Sym(96)| (1,41,35,56)(2,42,36,57)(3,43,25,58)(4,44,26,59)(5,45,27,60)(6,46,28,49)(7,47,29,50)(8,48,30,51)(9,37,31,52)(10,38,32,53)(11,39,33,54)(12,40,34,55)(13,63,96,78)(14,64,85,79)(15,65,86,80)(16,66,87,81)(17,67,88,82)(18,68,89,83)(19,69,90,84)(20,70,91,73)(21,71,92,74)(22,72,93,75)(23,61,94,76)(24,62,95,77), (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)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (1,68)(2,69)(3,70)(4,71)(5,72)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,49)(24,50)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)>;
G:=Group( (1,41,35,56)(2,42,36,57)(3,43,25,58)(4,44,26,59)(5,45,27,60)(6,46,28,49)(7,47,29,50)(8,48,30,51)(9,37,31,52)(10,38,32,53)(11,39,33,54)(12,40,34,55)(13,63,96,78)(14,64,85,79)(15,65,86,80)(16,66,87,81)(17,67,88,82)(18,68,89,83)(19,69,90,84)(20,70,91,73)(21,71,92,74)(22,72,93,75)(23,61,94,76)(24,62,95,77), (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)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (1,68)(2,69)(3,70)(4,71)(5,72)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,49)(24,50)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96) );
G=PermutationGroup([[(1,41,35,56),(2,42,36,57),(3,43,25,58),(4,44,26,59),(5,45,27,60),(6,46,28,49),(7,47,29,50),(8,48,30,51),(9,37,31,52),(10,38,32,53),(11,39,33,54),(12,40,34,55),(13,63,96,78),(14,64,85,79),(15,65,86,80),(16,66,87,81),(17,67,88,82),(18,68,89,83),(19,69,90,84),(20,70,91,73),(21,71,92,74),(22,72,93,75),(23,61,94,76),(24,62,95,77)], [(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),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,29,33),(26,30,34),(27,31,35),(28,32,36),(37,41,45),(38,42,46),(39,43,47),(40,44,48),(49,53,57),(50,54,58),(51,55,59),(52,56,60),(61,69,65),(62,70,66),(63,71,67),(64,72,68),(73,81,77),(74,82,78),(75,83,79),(76,84,80),(85,93,89),(86,94,90),(87,95,91),(88,96,92)], [(1,68),(2,69),(3,70),(4,71),(5,72),(6,61),(7,62),(8,63),(9,64),(10,65),(11,66),(12,67),(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,57),(20,58),(21,59),(22,60),(23,49),(24,50),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,88),(41,89),(42,90),(43,91),(44,92),(45,93),(46,94),(47,95),(48,96)]])
144 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 3E | 4A | ··· | 4L | 4M | ··· | 4X | 6A | ··· | 6F | 6G | ··· | 6O | 6P | ··· | 6W | 12A | ··· | 12X | 12Y | ··· | 12BH | 12BI | ··· | 12CF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
144 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×C4×C12 | Dic3×C12 | C122 | S3×C2×C12 | S3×C42 | S3×C12 | C4×Dic3 | C4×C12 | S3×C2×C4 | C4×S3 | C4×C12 | C2×C12 | C42 | C12 | C2×C4 | C4 |
| # reps | 1 | 3 | 1 | 3 | 2 | 24 | 6 | 2 | 6 | 48 | 1 | 3 | 2 | 12 | 6 | 24 |
Matrix representation of S3×C4×C12 ►in GL3(𝔽13) generated by
| 1 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 7 | 0 | 0 |
| 0 | 7 | 0 |
| 0 | 0 | 7 |
| 1 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 6 | 3 |
| 12 | 0 | 0 |
| 0 | 12 | 1 |
| 0 | 0 | 1 |
G:=sub<GL(3,GF(13))| [1,0,0,0,8,0,0,0,8],[7,0,0,0,7,0,0,0,7],[1,0,0,0,9,6,0,0,3],[12,0,0,0,12,0,0,1,1] >;
S3×C4×C12 in GAP, Magma, Sage, TeX
S_3\times C_4\times C_{12} % in TeX
G:=Group("S3xC4xC12"); // GroupNames label
G:=SmallGroup(288,642);
// by ID
G=gap.SmallGroup(288,642);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,142,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^4=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