metabelian, supersoluble, monomial
Aliases: D12:3Dic3, C12.15D12, C62.17D4, (C3xD12):1C4, (C3xC6).13D8, C12.13(C4xS3), C4:Dic3:10S3, (C6xD12).2C2, (C2xD12).6S3, (C2xC12).75D6, (C3xC12).32D4, (C3xC6).7SD16, C4.6(S3xDic3), C6.34(D6:C4), C6.11(D4:S3), C3:3(C6.D8), C6.4(D4.S3), C12.7(C2xDic3), C3:2(D4:Dic3), C12.28(C3:D4), C32:4(D4:C4), C2.5(D6:Dic3), (C6xC12).24C22, C6.4(Q8:2S3), C2.1(C32:2D8), C4.21(C3:D12), C6.4(C6.D4), C2.1(Dic6:S3), C22.7(D6:S3), (C2xC4).99S32, (C3xC4:Dic3):1C2, (C3xC12).22(C2xC4), (C2xC32:4C8):1C2, (C2xC6).48(C3:D4), (C3xC6).29(C22:C4), SmallGroup(288,210)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12:3Dic3
G = < a,b,c,d | a12=b2=c6=1, d2=c3, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a3b, dcd-1=c-1 >
Subgroups: 402 in 111 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, C23, C32, Dic3, C12, C12, D6, C2xC6, C2xC6, C4:C4, C2xC8, C2xD4, C3xS3, C3xC6, C3:C8, D12, D12, C2xDic3, C2xC12, C2xC12, C3xD4, C22xS3, C22xC6, D4:C4, C3xDic3, C3xC12, S3xC6, C62, C2xC3:C8, C4:Dic3, C3xC4:C4, C2xD12, C6xD4, C32:4C8, C3xD12, C3xD12, C6xDic3, C6xC12, S3xC2xC6, C6.D8, D4:Dic3, C3xC4:Dic3, C2xC32:4C8, C6xD12, D12:3Dic3
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, Dic3, D6, C22:C4, D8, SD16, C4xS3, D12, C2xDic3, C3:D4, D4:C4, S32, D6:C4, D4:S3, D4.S3, Q8:2S3, C6.D4, S3xDic3, D6:S3, C3:D12, C6.D8, D4:Dic3, C32:2D8, Dic6:S3, D6:Dic3, D12:3Dic3
►On 96 points
Generators in S96
(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 67)(2 66)(3 65)(4 64)(5 63)(6 62)(7 61)(8 72)(9 71)(10 70)(11 69)(12 68)(13 58)(14 57)(15 56)(16 55)(17 54)(18 53)(19 52)(20 51)(21 50)(22 49)(23 60)(24 59)(25 88)(26 87)(27 86)(28 85)(29 96)(30 95)(31 94)(32 93)(33 92)(34 91)(35 90)(36 89)(37 83)(38 82)(39 81)(40 80)(41 79)(42 78)(43 77)(44 76)(45 75)(46 74)(47 73)(48 84)
(1 17 9 13 5 21)(2 18 10 14 6 22)(3 19 11 15 7 23)(4 20 12 16 8 24)(25 45 29 37 33 41)(26 46 30 38 34 42)(27 47 31 39 35 43)(28 48 32 40 36 44)(49 66 53 70 57 62)(50 67 54 71 58 63)(51 68 55 72 59 64)(52 69 56 61 60 65)(73 94 81 90 77 86)(74 95 82 91 78 87)(75 96 83 92 79 88)(76 85 84 93 80 89)
(1 43 13 31)(2 38 14 26)(3 45 15 33)(4 40 16 28)(5 47 17 35)(6 42 18 30)(7 37 19 25)(8 44 20 32)(9 39 21 27)(10 46 22 34)(11 41 23 29)(12 48 24 36)(49 88 70 83)(50 95 71 78)(51 90 72 73)(52 85 61 80)(53 92 62 75)(54 87 63 82)(55 94 64 77)(56 89 65 84)(57 96 66 79)(58 91 67 74)(59 86 68 81)(60 93 69 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), (1,67)(2,66)(3,65)(4,64)(5,63)(6,62)(7,61)(8,72)(9,71)(10,70)(11,69)(12,68)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,60)(24,59)(25,88)(26,87)(27,86)(28,85)(29,96)(30,95)(31,94)(32,93)(33,92)(34,91)(35,90)(36,89)(37,83)(38,82)(39,81)(40,80)(41,79)(42,78)(43,77)(44,76)(45,75)(46,74)(47,73)(48,84), (1,17,9,13,5,21)(2,18,10,14,6,22)(3,19,11,15,7,23)(4,20,12,16,8,24)(25,45,29,37,33,41)(26,46,30,38,34,42)(27,47,31,39,35,43)(28,48,32,40,36,44)(49,66,53,70,57,62)(50,67,54,71,58,63)(51,68,55,72,59,64)(52,69,56,61,60,65)(73,94,81,90,77,86)(74,95,82,91,78,87)(75,96,83,92,79,88)(76,85,84,93,80,89), (1,43,13,31)(2,38,14,26)(3,45,15,33)(4,40,16,28)(5,47,17,35)(6,42,18,30)(7,37,19,25)(8,44,20,32)(9,39,21,27)(10,46,22,34)(11,41,23,29)(12,48,24,36)(49,88,70,83)(50,95,71,78)(51,90,72,73)(52,85,61,80)(53,92,62,75)(54,87,63,82)(55,94,64,77)(56,89,65,84)(57,96,66,79)(58,91,67,74)(59,86,68,81)(60,93,69,76)>;
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), (1,67)(2,66)(3,65)(4,64)(5,63)(6,62)(7,61)(8,72)(9,71)(10,70)(11,69)(12,68)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,60)(24,59)(25,88)(26,87)(27,86)(28,85)(29,96)(30,95)(31,94)(32,93)(33,92)(34,91)(35,90)(36,89)(37,83)(38,82)(39,81)(40,80)(41,79)(42,78)(43,77)(44,76)(45,75)(46,74)(47,73)(48,84), (1,17,9,13,5,21)(2,18,10,14,6,22)(3,19,11,15,7,23)(4,20,12,16,8,24)(25,45,29,37,33,41)(26,46,30,38,34,42)(27,47,31,39,35,43)(28,48,32,40,36,44)(49,66,53,70,57,62)(50,67,54,71,58,63)(51,68,55,72,59,64)(52,69,56,61,60,65)(73,94,81,90,77,86)(74,95,82,91,78,87)(75,96,83,92,79,88)(76,85,84,93,80,89), (1,43,13,31)(2,38,14,26)(3,45,15,33)(4,40,16,28)(5,47,17,35)(6,42,18,30)(7,37,19,25)(8,44,20,32)(9,39,21,27)(10,46,22,34)(11,41,23,29)(12,48,24,36)(49,88,70,83)(50,95,71,78)(51,90,72,73)(52,85,61,80)(53,92,62,75)(54,87,63,82)(55,94,64,77)(56,89,65,84)(57,96,66,79)(58,91,67,74)(59,86,68,81)(60,93,69,76) );
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)], [(1,67),(2,66),(3,65),(4,64),(5,63),(6,62),(7,61),(8,72),(9,71),(10,70),(11,69),(12,68),(13,58),(14,57),(15,56),(16,55),(17,54),(18,53),(19,52),(20,51),(21,50),(22,49),(23,60),(24,59),(25,88),(26,87),(27,86),(28,85),(29,96),(30,95),(31,94),(32,93),(33,92),(34,91),(35,90),(36,89),(37,83),(38,82),(39,81),(40,80),(41,79),(42,78),(43,77),(44,76),(45,75),(46,74),(47,73),(48,84)], [(1,17,9,13,5,21),(2,18,10,14,6,22),(3,19,11,15,7,23),(4,20,12,16,8,24),(25,45,29,37,33,41),(26,46,30,38,34,42),(27,47,31,39,35,43),(28,48,32,40,36,44),(49,66,53,70,57,62),(50,67,54,71,58,63),(51,68,55,72,59,64),(52,69,56,61,60,65),(73,94,81,90,77,86),(74,95,82,91,78,87),(75,96,83,92,79,88),(76,85,84,93,80,89)], [(1,43,13,31),(2,38,14,26),(3,45,15,33),(4,40,16,28),(5,47,17,35),(6,42,18,30),(7,37,19,25),(8,44,20,32),(9,39,21,27),(10,46,22,34),(11,41,23,29),(12,48,24,36),(49,88,70,83),(50,95,71,78),(51,90,72,73),(52,85,61,80),(53,92,62,75),(54,87,63,82),(55,94,64,77),(56,89,65,84),(57,96,66,79),(58,91,67,74),(59,86,68,81),(60,93,69,76)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 8A | 8B | 8C | 8D | 12A | ··· | 12H | 12I | 12J | 12K | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | 4 | 2 | 2 | 12 | 12 | 2 | ··· | 2 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 4 | ··· | 4 | 12 | 12 | 12 | 12 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | S3 | D4 | D4 | Dic3 | D6 | D8 | SD16 | C4xS3 | D12 | C3:D4 | C3:D4 | S32 | D4:S3 | D4.S3 | Q8:2S3 | S3xDic3 | C3:D12 | D6:S3 | C32:2D8 | Dic6:S3 |
| kernel | D12:3Dic3 | C3xC4:Dic3 | C2xC32:4C8 | C6xD12 | C3xD12 | C4:Dic3 | C2xD12 | C3xC12 | C62 | D12 | C2xC12 | C3xC6 | C3xC6 | C12 | C12 | C12 | C2xC6 | C2xC4 | C6 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of D12:3Dic3 ►in GL6(F73)
| 27 | 0 | 0 | 0 | 0 | 0 |
| 71 | 46 | 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 | 1 | 0 |
| 38 | 4 | 0 | 0 | 0 | 0 |
| 59 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 54 | 48 |
| 0 | 0 | 0 | 0 | 29 | 19 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 58 | 0 | 0 | 0 | 0 |
| 24 | 33 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 27 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 13 |
| 0 | 0 | 0 | 0 | 60 | 43 |
G:=sub<GL(6,GF(73))| [27,71,0,0,0,0,0,46,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,72,0],[38,59,0,0,0,0,4,35,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,54,29,0,0,0,0,48,19],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,24,0,0,0,0,58,33,0,0,0,0,0,0,46,27,0,0,0,0,0,27,0,0,0,0,0,0,30,60,0,0,0,0,13,43] >;
D12:3Dic3 in GAP, Magma, Sage, TeX
D_{12}\rtimes_3{\rm Dic}_3 % in TeX
G:=Group("D12:3Dic3"); // GroupNames label
G:=SmallGroup(288,210);
// by ID
G=gap.SmallGroup(288,210);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,675,346,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^6=1,d^2=c^3,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations