metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12:12D4, C42.172D6, C6.352- 1+4, C4:Q8:10S3, C4.73(S3xD4), (C4xD12):51C2, C12:7(C4oD4), C3:7(D4:6D4), C4:C4.123D6, D6.25(C2xD4), C12.71(C2xD4), C12:D4:40C2, D6:3Q8:35C2, C4:2(Q8:3S3), (C2xQ8).169D6, D6.D4:46C2, (C2xC6).270C24, C6.100(C22xD4), (C2xC12).103C23, (C4xC12).211C22, D6:C4.151C22, (C6xQ8).137C22, (C2xD12).271C22, Dic3:C4.60C22, C4:Dic3.384C22, C22.291(S3xC23), (C22xS3).231C23, C2.36(Q8.15D6), (C2xDic3).141C23, (S3xC4:C4):44C2, C2.73(C2xS3xD4), (C3xC4:Q8):12C2, C6.121(C2xC4oD4), (C2xQ8:3S3):13C2, (S3xC2xC4).144C22, (C2xC4).93(C22xS3), C2.28(C2xQ8:3S3), (C3xC4:C4).213C22, SmallGroup(192,1285)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12:12D4
G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, ac=ca, dad=a5, bc=cb, dbd=a10b, dcd=c-1 >
Subgroups: 768 in 292 conjugacy classes, 107 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2xC4, C2xC4, C2xC4, D4, Q8, C23, Dic3, C12, C12, D6, D6, C2xC6, C42, C22:C4, C4:C4, C4:C4, C22xC4, C2xD4, C2xQ8, C4oD4, C4xS3, D12, D12, C2xDic3, C2xC12, C2xC12, C3xQ8, C22xS3, C2xC4:C4, C4xD4, C4:D4, C22:Q8, C22.D4, C4:Q8, C2xC4oD4, Dic3:C4, C4:Dic3, D6:C4, C4xC12, C3xC4:C4, S3xC2xC4, C2xD12, C2xD12, Q8:3S3, C6xQ8, D4:6D4, C4xD12, S3xC4:C4, D6.D4, C12:D4, D6:3Q8, C3xC4:Q8, C2xQ8:3S3, D12:12D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C24, C22xS3, C22xD4, C2xC4oD4, 2- 1+4, S3xD4, Q8:3S3, S3xC23, D4:6D4, C2xS3xD4, C2xQ8:3S3, Q8.15D6, D12:12D4
►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 14)(2 13)(3 24)(4 23)(5 22)(6 21)(7 20)(8 19)(9 18)(10 17)(11 16)(12 15)(25 63)(26 62)(27 61)(28 72)(29 71)(30 70)(31 69)(32 68)(33 67)(34 66)(35 65)(36 64)(37 59)(38 58)(39 57)(40 56)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(48 60)(73 87)(74 86)(75 85)(76 96)(77 95)(78 94)(79 93)(80 92)(81 91)(82 90)(83 89)(84 88)
(1 73 48 68)(2 74 37 69)(3 75 38 70)(4 76 39 71)(5 77 40 72)(6 78 41 61)(7 79 42 62)(8 80 43 63)(9 81 44 64)(10 82 45 65)(11 83 46 66)(12 84 47 67)(13 86 59 31)(14 87 60 32)(15 88 49 33)(16 89 50 34)(17 90 51 35)(18 91 52 36)(19 92 53 25)(20 93 54 26)(21 94 55 27)(22 95 56 28)(23 96 57 29)(24 85 58 30)
(1 48)(2 41)(3 46)(4 39)(5 44)(6 37)(7 42)(8 47)(9 40)(10 45)(11 38)(12 43)(13 57)(14 50)(15 55)(16 60)(17 53)(18 58)(19 51)(20 56)(21 49)(22 54)(23 59)(24 52)(25 35)(26 28)(27 33)(29 31)(30 36)(32 34)(61 69)(63 67)(64 72)(66 70)(74 78)(75 83)(77 81)(80 84)(85 91)(86 96)(87 89)(88 94)(90 92)(93 95)
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), (1,14)(2,13)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(25,63)(26,62)(27,61)(28,72)(29,71)(30,70)(31,69)(32,68)(33,67)(34,66)(35,65)(36,64)(37,59)(38,58)(39,57)(40,56)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(48,60)(73,87)(74,86)(75,85)(76,96)(77,95)(78,94)(79,93)(80,92)(81,91)(82,90)(83,89)(84,88), (1,73,48,68)(2,74,37,69)(3,75,38,70)(4,76,39,71)(5,77,40,72)(6,78,41,61)(7,79,42,62)(8,80,43,63)(9,81,44,64)(10,82,45,65)(11,83,46,66)(12,84,47,67)(13,86,59,31)(14,87,60,32)(15,88,49,33)(16,89,50,34)(17,90,51,35)(18,91,52,36)(19,92,53,25)(20,93,54,26)(21,94,55,27)(22,95,56,28)(23,96,57,29)(24,85,58,30), (1,48)(2,41)(3,46)(4,39)(5,44)(6,37)(7,42)(8,47)(9,40)(10,45)(11,38)(12,43)(13,57)(14,50)(15,55)(16,60)(17,53)(18,58)(19,51)(20,56)(21,49)(22,54)(23,59)(24,52)(25,35)(26,28)(27,33)(29,31)(30,36)(32,34)(61,69)(63,67)(64,72)(66,70)(74,78)(75,83)(77,81)(80,84)(85,91)(86,96)(87,89)(88,94)(90,92)(93,95)>;
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), (1,14)(2,13)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(25,63)(26,62)(27,61)(28,72)(29,71)(30,70)(31,69)(32,68)(33,67)(34,66)(35,65)(36,64)(37,59)(38,58)(39,57)(40,56)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(48,60)(73,87)(74,86)(75,85)(76,96)(77,95)(78,94)(79,93)(80,92)(81,91)(82,90)(83,89)(84,88), (1,73,48,68)(2,74,37,69)(3,75,38,70)(4,76,39,71)(5,77,40,72)(6,78,41,61)(7,79,42,62)(8,80,43,63)(9,81,44,64)(10,82,45,65)(11,83,46,66)(12,84,47,67)(13,86,59,31)(14,87,60,32)(15,88,49,33)(16,89,50,34)(17,90,51,35)(18,91,52,36)(19,92,53,25)(20,93,54,26)(21,94,55,27)(22,95,56,28)(23,96,57,29)(24,85,58,30), (1,48)(2,41)(3,46)(4,39)(5,44)(6,37)(7,42)(8,47)(9,40)(10,45)(11,38)(12,43)(13,57)(14,50)(15,55)(16,60)(17,53)(18,58)(19,51)(20,56)(21,49)(22,54)(23,59)(24,52)(25,35)(26,28)(27,33)(29,31)(30,36)(32,34)(61,69)(63,67)(64,72)(66,70)(74,78)(75,83)(77,81)(80,84)(85,91)(86,96)(87,89)(88,94)(90,92)(93,95) );
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)], [(1,14),(2,13),(3,24),(4,23),(5,22),(6,21),(7,20),(8,19),(9,18),(10,17),(11,16),(12,15),(25,63),(26,62),(27,61),(28,72),(29,71),(30,70),(31,69),(32,68),(33,67),(34,66),(35,65),(36,64),(37,59),(38,58),(39,57),(40,56),(41,55),(42,54),(43,53),(44,52),(45,51),(46,50),(47,49),(48,60),(73,87),(74,86),(75,85),(76,96),(77,95),(78,94),(79,93),(80,92),(81,91),(82,90),(83,89),(84,88)], [(1,73,48,68),(2,74,37,69),(3,75,38,70),(4,76,39,71),(5,77,40,72),(6,78,41,61),(7,79,42,62),(8,80,43,63),(9,81,44,64),(10,82,45,65),(11,83,46,66),(12,84,47,67),(13,86,59,31),(14,87,60,32),(15,88,49,33),(16,89,50,34),(17,90,51,35),(18,91,52,36),(19,92,53,25),(20,93,54,26),(21,94,55,27),(22,95,56,28),(23,96,57,29),(24,85,58,30)], [(1,48),(2,41),(3,46),(4,39),(5,44),(6,37),(7,42),(8,47),(9,40),(10,45),(11,38),(12,43),(13,57),(14,50),(15,55),(16,60),(17,53),(18,58),(19,51),(20,56),(21,49),(22,54),(23,59),(24,52),(25,35),(26,28),(27,33),(29,31),(30,36),(32,34),(61,69),(63,67),(64,72),(66,70),(74,78),(75,83),(77,81),(80,84),(85,91),(86,96),(87,89),(88,94),(90,92),(93,95)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | C4oD4 | 2- 1+4 | S3xD4 | Q8:3S3 | Q8.15D6 |
| kernel | D12:12D4 | C4xD12 | S3xC4:C4 | D6.D4 | C12:D4 | D6:3Q8 | C3xC4:Q8 | C2xQ8:3S3 | C4:Q8 | D12 | C42 | C4:C4 | C2xQ8 | C12 | C6 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 4 | 2 | 2 | 1 | 2 | 1 | 4 | 1 | 4 | 2 | 4 | 1 | 2 | 2 | 2 |
Matrix representation of D12:12D4 ►in GL6(F13)
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 2 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 5 | 1 |
G:=sub<GL(6,GF(13))| [8,0,0,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,2,8],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,5,0,0,0,0,0,1] >;
D12:12D4 in GAP, Magma, Sage, TeX
D_{12}\rtimes_{12}D_4 % in TeX
G:=Group("D12:12D4"); // GroupNames label
G:=SmallGroup(192,1285);
// by ID
G=gap.SmallGroup(192,1285);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,268,1571,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^5,b*c=c*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations