direct product, metabelian, supersoluble, monomial
Aliases: C12×D12, C122⋊10C2, C62.164C23, C4⋊2(S3×C12), C12⋊5(C3×D4), C3⋊1(D4×C12), C6.2(C6×D4), C12⋊11(C4×S3), (C4×C12)⋊13S3, C12⋊4(C2×C12), (C4×C12)⋊13C6, D6⋊C4⋊17C6, D6⋊1(C2×C12), (C3×C12)⋊19D4, C42⋊8(C3×S3), C2.1(C6×D12), C32⋊16(C4×D4), C4⋊Dic3⋊16C6, C6.90(C2×D12), (C2×D12).10C6, (C6×D12).20C2, (C2×C12).459D6, C6.4(C22×C12), C6.113(C4○D12), (C6×C12).346C22, (C6×Dic3).120C22, (S3×C2×C4)⋊7C6, C2.6(S3×C2×C12), (S3×C2×C12)⋊22C2, C6.103(S3×C2×C4), C6.4(C3×C4○D4), (S3×C6)⋊14(C2×C4), (C3×C12)⋊18(C2×C4), (C3×D6⋊C4)⋊38C2, (C2×C4).97(S3×C6), C2.3(C3×C4○D12), C22.11(S3×C2×C6), (C3×C4⋊Dic3)⋊34C2, (C3×C6).173(C2×D4), (S3×C2×C6).87C22, (C2×C12).127(C2×C6), (C3×C6).94(C4○D4), (C2×C6).19(C22×C6), (C3×C6).75(C22×C4), (C22×S3).15(C2×C6), (C2×C6).297(C22×S3), (C2×Dic3).17(C2×C6), SmallGroup(288,644)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12×D12
G = < a,b,c | a12=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 466 in 203 conjugacy classes, 90 normal (42 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3×C6, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C4×D4, C3×Dic3, C3×C12, C3×C12, S3×C6, S3×C6, C62, C4⋊Dic3, D6⋊C4, C4×C12, C4×C12, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C22×C12, C6×D4, S3×C12, C3×D12, C6×Dic3, C6×C12, S3×C2×C6, C4×D12, D4×C12, C3×C4⋊Dic3, C3×D6⋊C4, C122, S3×C2×C12, C6×D12, C12×D12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, C23, C12, D6, C2×C6, C22×C4, C2×D4, C4○D4, C3×S3, C4×S3, D12, C2×C12, C3×D4, C22×S3, C22×C6, C4×D4, S3×C6, S3×C2×C4, C2×D12, C4○D12, C22×C12, C6×D4, C3×C4○D4, S3×C12, C3×D12, S3×C2×C6, C4×D12, D4×C12, S3×C2×C12, C6×D12, C3×C4○D12, C12×D12
►On 96 points
(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 90 15 31 5 94 19 35 9 86 23 27)(2 91 16 32 6 95 20 36 10 87 24 28)(3 92 17 33 7 96 21 25 11 88 13 29)(4 93 18 34 8 85 22 26 12 89 14 30)(37 84 52 67 45 80 60 63 41 76 56 71)(38 73 53 68 46 81 49 64 42 77 57 72)(39 74 54 69 47 82 50 65 43 78 58 61)(40 75 55 70 48 83 51 66 44 79 59 62)
(1 71)(2 72)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 80)(20 81)(21 82)(22 83)(23 84)(24 73)(25 47)(26 48)(27 37)(28 38)(29 39)(30 40)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)(49 95)(50 96)(51 85)(52 86)(53 87)(54 88)(55 89)(56 90)(57 91)(58 92)(59 93)(60 94)
G:=sub<Sym(96)| (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), (1,90,15,31,5,94,19,35,9,86,23,27)(2,91,16,32,6,95,20,36,10,87,24,28)(3,92,17,33,7,96,21,25,11,88,13,29)(4,93,18,34,8,85,22,26,12,89,14,30)(37,84,52,67,45,80,60,63,41,76,56,71)(38,73,53,68,46,81,49,64,42,77,57,72)(39,74,54,69,47,82,50,65,43,78,58,61)(40,75,55,70,48,83,51,66,44,79,59,62), (1,71)(2,72)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,73)(25,47)(26,48)(27,37)(28,38)(29,39)(30,40)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(49,95)(50,96)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,91)(58,92)(59,93)(60,94)>;
G:=Group( (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), (1,90,15,31,5,94,19,35,9,86,23,27)(2,91,16,32,6,95,20,36,10,87,24,28)(3,92,17,33,7,96,21,25,11,88,13,29)(4,93,18,34,8,85,22,26,12,89,14,30)(37,84,52,67,45,80,60,63,41,76,56,71)(38,73,53,68,46,81,49,64,42,77,57,72)(39,74,54,69,47,82,50,65,43,78,58,61)(40,75,55,70,48,83,51,66,44,79,59,62), (1,71)(2,72)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,73)(25,47)(26,48)(27,37)(28,38)(29,39)(30,40)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(49,95)(50,96)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,91)(58,92)(59,93)(60,94) );
G=PermutationGroup([[(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)], [(1,90,15,31,5,94,19,35,9,86,23,27),(2,91,16,32,6,95,20,36,10,87,24,28),(3,92,17,33,7,96,21,25,11,88,13,29),(4,93,18,34,8,85,22,26,12,89,14,30),(37,84,52,67,45,80,60,63,41,76,56,71),(38,73,53,68,46,81,49,64,42,77,57,72),(39,74,54,69,47,82,50,65,43,78,58,61),(40,75,55,70,48,83,51,66,44,79,59,62)], [(1,71),(2,72),(3,61),(4,62),(5,63),(6,64),(7,65),(8,66),(9,67),(10,68),(11,69),(12,70),(13,74),(14,75),(15,76),(16,77),(17,78),(18,79),(19,80),(20,81),(21,82),(22,83),(23,84),(24,73),(25,47),(26,48),(27,37),(28,38),(29,39),(30,40),(31,41),(32,42),(33,43),(34,44),(35,45),(36,46),(49,95),(50,96),(51,85),(52,86),(53,87),(54,88),(55,89),(56,90),(57,91),(58,92),(59,93),(60,94)]])
108 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | ··· | 6F | 6G | ··· | 6O | 6P | ··· | 6W | 12A | ··· | 12H | 12I | ··· | 12AZ | 12BA | ··· | 12BH |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | S3 | D4 | D6 | C4○D4 | C3×S3 | C4×S3 | D12 | C3×D4 | S3×C6 | C4○D12 | C3×C4○D4 | S3×C12 | C3×D12 | C3×C4○D12 |
| kernel | C12×D12 | C3×C4⋊Dic3 | C3×D6⋊C4 | C122 | S3×C2×C12 | C6×D12 | C4×D12 | C3×D12 | C4⋊Dic3 | D6⋊C4 | C4×C12 | S3×C2×C4 | C2×D12 | D12 | C4×C12 | C3×C12 | C2×C12 | C3×C6 | C42 | C12 | C12 | C12 | C2×C4 | C6 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 8 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 2 | 3 | 2 | 2 | 4 | 4 | 4 | 6 | 4 | 4 | 8 | 8 | 8 |
Matrix representation of C12×D12 ►in GL4(𝔽13) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 8 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 11 |
| 0 | 5 | 0 | 0 |
| 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 |
| 0 | 0 | 6 | 0 |
G:=sub<GL(4,GF(13))| [8,0,0,0,0,8,0,0,0,0,9,0,0,0,0,9],[8,0,0,0,0,5,0,0,0,0,6,0,0,0,0,11],[0,8,0,0,5,0,0,0,0,0,0,6,0,0,11,0] >;
C12×D12 in GAP, Magma, Sage, TeX
C_{12}\times D_{12} % in TeX
G:=Group("C12xD12"); // GroupNames label
G:=SmallGroup(288,644);
// by ID
G=gap.SmallGroup(288,644);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,344,142,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations