direct product, metabelian, supersoluble, monomial
Aliases: C3xC4:D12, C12:6D12, C122:11C2, C62.165C23, C12:4(C3xD4), C4:1(C3xD12), C6.3(C6xD4), (C4xC12):14S3, (C4xC12):10C6, (C2xD12):1C6, (C3xC12):18D4, C42:9(C3xS3), C2.5(C6xD12), (C6xD12):24C2, C6.91(C2xD12), (C2xC12).439D6, C32:5(C4:1D4), (C6xC12).320C22, C3:1(C3xC4:1D4), (C2xC4).76(S3xC6), C22.36(S3xC2xC6), (C3xC6).174(C2xD4), (S3xC2xC6).52C22, (C2xC12).105(C2xC6), (C22xS3).1(C2xC6), (C2xC6).20(C22xC6), (C2xC6).298(C22xS3), SmallGroup(288,645)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3xC4:D12
G = < a,b,c,d | a3=b4=c12=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 690 in 231 conjugacy classes, 82 normal (14 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, D4, C23, C32, C12, C12, D6, C2xC6, C2xC6, C42, C2xD4, C3xS3, C3xC6, D12, C2xC12, C2xC12, C3xD4, C22xS3, C22xC6, C4:1D4, C3xC12, S3xC6, C62, C4xC12, C4xC12, C2xD12, C6xD4, C3xD12, C6xC12, S3xC2xC6, C4:D12, C3xC4:1D4, C122, C6xD12, C3xC4:D12
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2xC6, C2xD4, C3xS3, D12, C3xD4, C22xS3, C22xC6, C4:1D4, S3xC6, C2xD12, C6xD4, C3xD12, S3xC2xC6, C4:D12, C3xC4:1D4, C6xD12, C3xC4:D12
►On 96 points
Generators in S96
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)(49 57 53)(50 58 54)(51 59 55)(52 60 56)(61 69 65)(62 70 66)(63 71 67)(64 72 68)(73 77 81)(74 78 82)(75 79 83)(76 80 84)(85 89 93)(86 90 94)(87 91 95)(88 92 96)
(1 71 46 55)(2 72 47 56)(3 61 48 57)(4 62 37 58)(5 63 38 59)(6 64 39 60)(7 65 40 49)(8 66 41 50)(9 67 42 51)(10 68 43 52)(11 69 44 53)(12 70 45 54)(13 88 29 83)(14 89 30 84)(15 90 31 73)(16 91 32 74)(17 92 33 75)(18 93 34 76)(19 94 35 77)(20 95 36 78)(21 96 25 79)(22 85 26 80)(23 86 27 81)(24 87 28 82)
(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 82)(2 81)(3 80)(4 79)(5 78)(6 77)(7 76)(8 75)(9 74)(10 73)(11 84)(12 83)(13 54)(14 53)(15 52)(16 51)(17 50)(18 49)(19 60)(20 59)(21 58)(22 57)(23 56)(24 55)(25 62)(26 61)(27 72)(28 71)(29 70)(30 69)(31 68)(32 67)(33 66)(34 65)(35 64)(36 63)(37 96)(38 95)(39 94)(40 93)(41 92)(42 91)(43 90)(44 89)(45 88)(46 87)(47 86)(48 85)
G:=sub<Sym(96)| (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,77,81)(74,78,82)(75,79,83)(76,80,84)(85,89,93)(86,90,94)(87,91,95)(88,92,96), (1,71,46,55)(2,72,47,56)(3,61,48,57)(4,62,37,58)(5,63,38,59)(6,64,39,60)(7,65,40,49)(8,66,41,50)(9,67,42,51)(10,68,43,52)(11,69,44,53)(12,70,45,54)(13,88,29,83)(14,89,30,84)(15,90,31,73)(16,91,32,74)(17,92,33,75)(18,93,34,76)(19,94,35,77)(20,95,36,78)(21,96,25,79)(22,85,26,80)(23,86,27,81)(24,87,28,82), (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), (1,82)(2,81)(3,80)(4,79)(5,78)(6,77)(7,76)(8,75)(9,74)(10,73)(11,84)(12,83)(13,54)(14,53)(15,52)(16,51)(17,50)(18,49)(19,60)(20,59)(21,58)(22,57)(23,56)(24,55)(25,62)(26,61)(27,72)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,96)(38,95)(39,94)(40,93)(41,92)(42,91)(43,90)(44,89)(45,88)(46,87)(47,86)(48,85)>;
G:=Group( (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,77,81)(74,78,82)(75,79,83)(76,80,84)(85,89,93)(86,90,94)(87,91,95)(88,92,96), (1,71,46,55)(2,72,47,56)(3,61,48,57)(4,62,37,58)(5,63,38,59)(6,64,39,60)(7,65,40,49)(8,66,41,50)(9,67,42,51)(10,68,43,52)(11,69,44,53)(12,70,45,54)(13,88,29,83)(14,89,30,84)(15,90,31,73)(16,91,32,74)(17,92,33,75)(18,93,34,76)(19,94,35,77)(20,95,36,78)(21,96,25,79)(22,85,26,80)(23,86,27,81)(24,87,28,82), (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), (1,82)(2,81)(3,80)(4,79)(5,78)(6,77)(7,76)(8,75)(9,74)(10,73)(11,84)(12,83)(13,54)(14,53)(15,52)(16,51)(17,50)(18,49)(19,60)(20,59)(21,58)(22,57)(23,56)(24,55)(25,62)(26,61)(27,72)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,96)(38,95)(39,94)(40,93)(41,92)(42,91)(43,90)(44,89)(45,88)(46,87)(47,86)(48,85) );
G=PermutationGroup([[(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,29,33),(26,30,34),(27,31,35),(28,32,36),(37,45,41),(38,46,42),(39,47,43),(40,48,44),(49,57,53),(50,58,54),(51,59,55),(52,60,56),(61,69,65),(62,70,66),(63,71,67),(64,72,68),(73,77,81),(74,78,82),(75,79,83),(76,80,84),(85,89,93),(86,90,94),(87,91,95),(88,92,96)], [(1,71,46,55),(2,72,47,56),(3,61,48,57),(4,62,37,58),(5,63,38,59),(6,64,39,60),(7,65,40,49),(8,66,41,50),(9,67,42,51),(10,68,43,52),(11,69,44,53),(12,70,45,54),(13,88,29,83),(14,89,30,84),(15,90,31,73),(16,91,32,74),(17,92,33,75),(18,93,34,76),(19,94,35,77),(20,95,36,78),(21,96,25,79),(22,85,26,80),(23,86,27,81),(24,87,28,82)], [(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)], [(1,82),(2,81),(3,80),(4,79),(5,78),(6,77),(7,76),(8,75),(9,74),(10,73),(11,84),(12,83),(13,54),(14,53),(15,52),(16,51),(17,50),(18,49),(19,60),(20,59),(21,58),(22,57),(23,56),(24,55),(25,62),(26,61),(27,72),(28,71),(29,70),(30,69),(31,68),(32,67),(33,66),(34,65),(35,64),(36,63),(37,96),(38,95),(39,94),(40,93),(41,92),(42,91),(43,90),(44,89),(45,88),(46,87),(47,86),(48,85)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 3E | 4A | ··· | 4F | 6A | ··· | 6F | 6G | ··· | 6O | 6P | ··· | 6W | 12A | ··· | 12AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 12 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | D4 | D6 | C3xS3 | D12 | C3xD4 | S3xC6 | C3xD12 |
| kernel | C3xC4:D12 | C122 | C6xD12 | C4:D12 | C4xC12 | C2xD12 | C4xC12 | C3xC12 | C2xC12 | C42 | C12 | C12 | C2xC4 | C4 |
| # reps | 1 | 1 | 6 | 2 | 2 | 12 | 1 | 6 | 3 | 2 | 12 | 12 | 6 | 24 |
Matrix representation of C3xC4:D12 ►in GL4(F13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 5 |
| 11 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 6 | 0 | 0 |
| 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 10 | 0 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,5],[11,0,0,0,0,6,0,0,0,0,10,0,0,0,0,4],[0,11,0,0,6,0,0,0,0,0,0,10,0,0,4,0] >;
C3xC4:D12 in GAP, Magma, Sage, TeX
C_3\times C_4\rtimes D_{12} % in TeX
G:=Group("C3xC4:D12"); // GroupNames label
G:=SmallGroup(288,645);
// by ID
G=gap.SmallGroup(288,645);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,344,590,142,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^12=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations