direct product, metabelian, supersoluble, monomial
Aliases: S3×Dic12, C24.22D6, D6.12D12, Dic3.3D12, Dic6.14D6, C8.8S32, C3⋊C8.23D6, C3⋊1(S3×Q16), C6.5(S3×D4), (S3×C8).1S3, (C3×S3)⋊1Q16, C6.5(C2×D12), (S3×C24).1C2, C32⋊3(C2×Q16), (S3×C6).19D4, (C4×S3).35D6, C3⋊1(C2×Dic12), C2.10(S3×D12), C32⋊5Q16⋊6C2, (C3×Dic12)⋊8C2, (S3×Dic6).1C2, C32⋊3Q16⋊1C2, (C3×C12).47C23, (C3×C24).21C22, (C3×Dic3).22D4, (S3×C12).43C22, C12.121(C22×S3), (C3×Dic6).1C22, C32⋊4Q8.1C22, C4.44(C2×S32), (C3×C6).31(C2×D4), (C3×C3⋊C8).28C22, SmallGroup(288,447)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×Dic12
G = < a,b,c,d | a3=b2=c24=1, d2=c12, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 498 in 129 conjugacy classes, 44 normal (30 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, Q8, C32, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C8, Q16, C2×Q8, C3×S3, C3×C6, C3⋊C8, C24, C24, Dic6, Dic6, C4×S3, C4×S3, C2×Dic3, C2×C12, C3×Q8, C2×Q16, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C8, Dic12, Dic12, C3⋊Q16, C2×C24, C3×Q16, C2×Dic6, S3×Q8, C3×C3⋊C8, C3×C24, S3×Dic3, C32⋊2Q8, C3×Dic6, S3×C12, C32⋊4Q8, C2×Dic12, S3×Q16, C32⋊3Q16, S3×C24, C3×Dic12, C32⋊5Q16, S3×Dic6, S3×Dic12
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, D12, C22×S3, C2×Q16, S32, Dic12, C2×D12, S3×D4, C2×S32, C2×Dic12, S3×Q16, S3×D12, S3×Dic12
►On 96 points
Generators in S96
(1 17 9)(2 18 10)(3 19 11)(4 20 12)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(25 33 41)(26 34 42)(27 35 43)(28 36 44)(29 37 45)(30 38 46)(31 39 47)(32 40 48)(49 57 65)(50 58 66)(51 59 67)(52 60 68)(53 61 69)(54 62 70)(55 63 71)(56 64 72)(73 89 81)(74 90 82)(75 91 83)(76 92 84)(77 93 85)(78 94 86)(79 95 87)(80 96 88)
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 93)(26 94)(27 95)(28 96)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 81)(38 82)(39 83)(40 84)(41 85)(42 86)(43 87)(44 88)(45 89)(46 90)(47 91)(48 92)
(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 37 13 25)(2 36 14 48)(3 35 15 47)(4 34 16 46)(5 33 17 45)(6 32 18 44)(7 31 19 43)(8 30 20 42)(9 29 21 41)(10 28 22 40)(11 27 23 39)(12 26 24 38)(49 93 61 81)(50 92 62 80)(51 91 63 79)(52 90 64 78)(53 89 65 77)(54 88 66 76)(55 87 67 75)(56 86 68 74)(57 85 69 73)(58 84 70 96)(59 83 71 95)(60 82 72 94)
G:=sub<Sym(96)| (1,17,9)(2,18,10)(3,19,11)(4,20,12)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(25,33,41)(26,34,42)(27,35,43)(28,36,44)(29,37,45)(30,38,46)(31,39,47)(32,40,48)(49,57,65)(50,58,66)(51,59,67)(52,60,68)(53,61,69)(54,62,70)(55,63,71)(56,64,72)(73,89,81)(74,90,82)(75,91,83)(76,92,84)(77,93,85)(78,94,86)(79,95,87)(80,96,88), (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,93)(26,94)(27,95)(28,96)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,81)(38,82)(39,83)(40,84)(41,85)(42,86)(43,87)(44,88)(45,89)(46,90)(47,91)(48,92), (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), (1,37,13,25)(2,36,14,48)(3,35,15,47)(4,34,16,46)(5,33,17,45)(6,32,18,44)(7,31,19,43)(8,30,20,42)(9,29,21,41)(10,28,22,40)(11,27,23,39)(12,26,24,38)(49,93,61,81)(50,92,62,80)(51,91,63,79)(52,90,64,78)(53,89,65,77)(54,88,66,76)(55,87,67,75)(56,86,68,74)(57,85,69,73)(58,84,70,96)(59,83,71,95)(60,82,72,94)>;
G:=Group( (1,17,9)(2,18,10)(3,19,11)(4,20,12)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(25,33,41)(26,34,42)(27,35,43)(28,36,44)(29,37,45)(30,38,46)(31,39,47)(32,40,48)(49,57,65)(50,58,66)(51,59,67)(52,60,68)(53,61,69)(54,62,70)(55,63,71)(56,64,72)(73,89,81)(74,90,82)(75,91,83)(76,92,84)(77,93,85)(78,94,86)(79,95,87)(80,96,88), (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,93)(26,94)(27,95)(28,96)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,81)(38,82)(39,83)(40,84)(41,85)(42,86)(43,87)(44,88)(45,89)(46,90)(47,91)(48,92), (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), (1,37,13,25)(2,36,14,48)(3,35,15,47)(4,34,16,46)(5,33,17,45)(6,32,18,44)(7,31,19,43)(8,30,20,42)(9,29,21,41)(10,28,22,40)(11,27,23,39)(12,26,24,38)(49,93,61,81)(50,92,62,80)(51,91,63,79)(52,90,64,78)(53,89,65,77)(54,88,66,76)(55,87,67,75)(56,86,68,74)(57,85,69,73)(58,84,70,96)(59,83,71,95)(60,82,72,94) );
G=PermutationGroup([[(1,17,9),(2,18,10),(3,19,11),(4,20,12),(5,21,13),(6,22,14),(7,23,15),(8,24,16),(25,33,41),(26,34,42),(27,35,43),(28,36,44),(29,37,45),(30,38,46),(31,39,47),(32,40,48),(49,57,65),(50,58,66),(51,59,67),(52,60,68),(53,61,69),(54,62,70),(55,63,71),(56,64,72),(73,89,81),(74,90,82),(75,91,83),(76,92,84),(77,93,85),(78,94,86),(79,95,87),(80,96,88)], [(1,61),(2,62),(3,63),(4,64),(5,65),(6,66),(7,67),(8,68),(9,69),(10,70),(11,71),(12,72),(13,49),(14,50),(15,51),(16,52),(17,53),(18,54),(19,55),(20,56),(21,57),(22,58),(23,59),(24,60),(25,93),(26,94),(27,95),(28,96),(29,73),(30,74),(31,75),(32,76),(33,77),(34,78),(35,79),(36,80),(37,81),(38,82),(39,83),(40,84),(41,85),(42,86),(43,87),(44,88),(45,89),(46,90),(47,91),(48,92)], [(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)], [(1,37,13,25),(2,36,14,48),(3,35,15,47),(4,34,16,46),(5,33,17,45),(6,32,18,44),(7,31,19,43),(8,30,20,42),(9,29,21,41),(10,28,22,40),(11,27,23,39),(12,26,24,38),(49,93,61,81),(50,92,62,80),(51,91,63,79),(52,90,64,78),(53,89,65,77),(54,88,66,76),(55,87,67,75),(56,86,68,74),(57,85,69,73),(58,84,70,96),(59,83,71,95),(60,82,72,94)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 24A | 24B | 24C | 24D | 24E | ··· | 24J | 24K | 24L | 24M | 24N |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 | 24 | ··· | 24 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 3 | 3 | 2 | 2 | 4 | 2 | 6 | 12 | 12 | 36 | 36 | 2 | 2 | 4 | 6 | 6 | 2 | 2 | 6 | 6 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 24 | 24 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | + | + | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D4 | D6 | D6 | D6 | D6 | Q16 | D12 | D12 | Dic12 | S32 | S3×D4 | C2×S32 | S3×Q16 | S3×D12 | S3×Dic12 |
| kernel | S3×Dic12 | C32⋊3Q16 | S3×C24 | C3×Dic12 | C32⋊5Q16 | S3×Dic6 | S3×C8 | Dic12 | C3×Dic3 | S3×C6 | C3⋊C8 | C24 | Dic6 | C4×S3 | C3×S3 | Dic3 | D6 | S3 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 8 | 1 | 1 | 1 | 2 | 2 | 4 |
Matrix representation of S3×Dic12 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 |
| 0 | 41 | 0 | 0 | 0 | 0 |
| 16 | 41 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 66 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 36 | 33 | 0 | 0 | 0 | 0 |
| 16 | 37 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 59 | 0 | 0 |
| 0 | 0 | 66 | 66 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[0,16,0,0,0,0,41,41,0,0,0,0,0,0,7,66,0,0,0,0,7,14,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,16,0,0,0,0,33,37,0,0,0,0,0,0,7,66,0,0,0,0,59,66,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
S3×Dic12 in GAP, Magma, Sage, TeX
S_3\times {\rm Dic}_{12} % in TeX
G:=Group("S3xDic12"); // GroupNames label
G:=SmallGroup(288,447);
// by ID
G=gap.SmallGroup(288,447);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,135,142,346,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^24=1,d^2=c^12,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations