metabelian, supersoluble, monomial
Aliases: C12:1Dic6, C62.89C23, (C3xC12):4Q8, C3:3(C12:Q8), C3:Dic3:7Q8, C6.44(S3xD4), C32:8(C4:Q8), C6.13(S3xQ8), (C2xC12).143D6, C4:2(C32:2Q8), C4:Dic3.13S3, C3:Dic3.45D4, C6.23(C2xDic6), (C2xDic3).37D6, C2.19(D6:D6), (C6xC12).111C22, C62.C22.6C2, (C6xDic3).21C22, C2.13(Dic3.D6), (C2xC4).121S32, (C3xC6).60(C2xD4), (C3xC6).35(C2xQ8), C22.126(C2xS32), (C4xC3:Dic3).4C2, C2.7(C2xC32:2Q8), (C3xC4:Dic3).17C2, (C2xC32:2Q8).4C2, (C2xC6).108(C22xS3), (C2xC3:Dic3).141C22, SmallGroup(288,567)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12:Dic6
G = < a,b,c | a12=b12=1, c2=b6, bab-1=a-1, cac-1=a7, cbc-1=b-1 >
Subgroups: 522 in 151 conjugacy classes, 56 normal (16 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C2xC4, C2xC4, Q8, C32, Dic3, C12, C12, C2xC6, C2xC6, C42, C4:C4, C2xQ8, C3xC6, Dic6, C2xDic3, C2xDic3, C2xC12, C2xC12, C4:Q8, C3xDic3, C3:Dic3, C3xC12, C62, C4xDic3, Dic3:C4, C4:Dic3, C3xC4:C4, C2xDic6, C32:2Q8, C6xDic3, C2xC3:Dic3, C6xC12, C12:Q8, C62.C22, C3xC4:Dic3, C4xC3:Dic3, C2xC32:2Q8, C12:Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2xD4, C2xQ8, Dic6, C22xS3, C4:Q8, S32, C2xDic6, S3xD4, S3xQ8, C32:2Q8, C2xS32, C12:Q8, Dic3.D6, D6:D6, C2xC32:2Q8, C12:Dic6
►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 91 46 21 9 95 42 13 5 87 38 17)(2 90 47 20 10 94 43 24 6 86 39 16)(3 89 48 19 11 93 44 23 7 85 40 15)(4 88 37 18 12 92 45 22 8 96 41 14)(25 50 80 72 33 54 76 64 29 58 84 68)(26 49 81 71 34 53 77 63 30 57 73 67)(27 60 82 70 35 52 78 62 31 56 74 66)(28 59 83 69 36 51 79 61 32 55 75 65)
(1 69 42 55)(2 64 43 50)(3 71 44 57)(4 66 45 52)(5 61 46 59)(6 68 47 54)(7 63 48 49)(8 70 37 56)(9 65 38 51)(10 72 39 58)(11 67 40 53)(12 62 41 60)(13 32 91 83)(14 27 92 78)(15 34 93 73)(16 29 94 80)(17 36 95 75)(18 31 96 82)(19 26 85 77)(20 33 86 84)(21 28 87 79)(22 35 88 74)(23 30 89 81)(24 25 90 76)
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), (1,91,46,21,9,95,42,13,5,87,38,17)(2,90,47,20,10,94,43,24,6,86,39,16)(3,89,48,19,11,93,44,23,7,85,40,15)(4,88,37,18,12,92,45,22,8,96,41,14)(25,50,80,72,33,54,76,64,29,58,84,68)(26,49,81,71,34,53,77,63,30,57,73,67)(27,60,82,70,35,52,78,62,31,56,74,66)(28,59,83,69,36,51,79,61,32,55,75,65), (1,69,42,55)(2,64,43,50)(3,71,44,57)(4,66,45,52)(5,61,46,59)(6,68,47,54)(7,63,48,49)(8,70,37,56)(9,65,38,51)(10,72,39,58)(11,67,40,53)(12,62,41,60)(13,32,91,83)(14,27,92,78)(15,34,93,73)(16,29,94,80)(17,36,95,75)(18,31,96,82)(19,26,85,77)(20,33,86,84)(21,28,87,79)(22,35,88,74)(23,30,89,81)(24,25,90,76)>;
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), (1,91,46,21,9,95,42,13,5,87,38,17)(2,90,47,20,10,94,43,24,6,86,39,16)(3,89,48,19,11,93,44,23,7,85,40,15)(4,88,37,18,12,92,45,22,8,96,41,14)(25,50,80,72,33,54,76,64,29,58,84,68)(26,49,81,71,34,53,77,63,30,57,73,67)(27,60,82,70,35,52,78,62,31,56,74,66)(28,59,83,69,36,51,79,61,32,55,75,65), (1,69,42,55)(2,64,43,50)(3,71,44,57)(4,66,45,52)(5,61,46,59)(6,68,47,54)(7,63,48,49)(8,70,37,56)(9,65,38,51)(10,72,39,58)(11,67,40,53)(12,62,41,60)(13,32,91,83)(14,27,92,78)(15,34,93,73)(16,29,94,80)(17,36,95,75)(18,31,96,82)(19,26,85,77)(20,33,86,84)(21,28,87,79)(22,35,88,74)(23,30,89,81)(24,25,90,76) );
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)], [(1,91,46,21,9,95,42,13,5,87,38,17),(2,90,47,20,10,94,43,24,6,86,39,16),(3,89,48,19,11,93,44,23,7,85,40,15),(4,88,37,18,12,92,45,22,8,96,41,14),(25,50,80,72,33,54,76,64,29,58,84,68),(26,49,81,71,34,53,77,63,30,57,73,67),(27,60,82,70,35,52,78,62,31,56,74,66),(28,59,83,69,36,51,79,61,32,55,75,65)], [(1,69,42,55),(2,64,43,50),(3,71,44,57),(4,66,45,52),(5,61,46,59),(6,68,47,54),(7,63,48,49),(8,70,37,56),(9,65,38,51),(10,72,39,58),(11,67,40,53),(12,62,41,60),(13,32,91,83),(14,27,92,78),(15,34,93,73),(16,29,94,80),(17,36,95,75),(18,31,96,82),(19,26,85,77),(20,33,86,84),(21,28,87,79),(22,35,88,74),(23,30,89,81),(24,25,90,76)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6F | 6G | 6H | 6I | 12A | ··· | 12H | 12I | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 12 | ··· | 12 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | + | + | - | + | + | - | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | Q8 | D6 | D6 | Dic6 | S32 | S3xD4 | S3xQ8 | C32:2Q8 | C2xS32 | Dic3.D6 | D6:D6 |
| kernel | C12:Dic6 | C62.C22 | C3xC4:Dic3 | C4xC3:Dic3 | C2xC32:2Q8 | C4:Dic3 | C3:Dic3 | C3:Dic3 | C3xC12 | C2xDic3 | C2xC12 | C12 | C2xC4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 8 | 1 | 2 | 2 | 2 | 1 | 2 | 2 |
Matrix representation of C12:Dic6 ►in GL6(F13)
| 8 | 6 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 2 | 5 | 0 | 0 | 0 | 0 |
| 12 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 10 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 10 | 2 | 0 | 0 | 0 | 0 |
| 8 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 8 | 0 |
G:=sub<GL(6,GF(13))| [8,0,0,0,0,0,6,5,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,12,0,0,0,0,5,11,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,6,3,0,0,0,0,10,3],[10,8,0,0,0,0,2,3,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,8,0] >;
C12:Dic6 in GAP, Magma, Sage, TeX
C_{12}\rtimes {\rm Dic}_6 % in TeX
G:=Group("C12:Dic6"); // GroupNames label
G:=SmallGroup(288,567);
// by ID
G=gap.SmallGroup(288,567);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,141,64,422,219,100,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=b^12=1,c^2=b^6,b*a*b^-1=a^-1,c*a*c^-1=a^7,c*b*c^-1=b^-1>;
// generators/relations