metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.1D12, C42.50D6, (C4×D4)⋊4S3, (D4×C12)⋊4C2, C4⋊C4.244D6, C12⋊C8⋊24C2, (C3×D4).18D4, (C2×C12).61D4, C4.14(C2×D12), C12.18(C2×D4), (C2×D4).191D6, C6.89(C4○D8), C6.D8⋊31C2, C3⋊4(D4.2D4), C42⋊7S3⋊13C2, C12.51(C4○D4), C4.10(C4○D12), C6.SD16⋊29C2, C6.65(C4⋊D4), C6.87(C8⋊C22), (C4×C12).87C22, C2.13(C12⋊7D4), (C2×C12).338C23, C2.9(D12⋊6C22), (C2×D12).94C22, (C6×D4).233C22, C2.11(Q8.13D6), (C2×Dic6).99C22, (C2×D4.S3)⋊7C2, (C2×D4⋊S3).5C2, (C2×C6).469(C2×D4), (C2×C3⋊C8).94C22, (C2×C4).218(C3⋊D4), (C3×C4⋊C4).275C22, (C2×C4).438(C22×S3), C22.150(C2×C3⋊D4), SmallGroup(192,575)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.1D12
G = < a,b,c,d | a4=b2=c12=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2c-1 >
Subgroups: 376 in 124 conjugacy classes, 43 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C3⋊C8, Dic6, D12, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C22×S3, C22×C6, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C2×C3⋊C8, D6⋊C4, D4⋊S3, D4.S3, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C2×D12, C22×C12, C6×D4, D4.2D4, C12⋊C8, C6.D8, C6.SD16, C42⋊7S3, C2×D4⋊S3, C2×D4.S3, D4×C12, D4.1D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C3⋊D4, C22×S3, C4⋊D4, C4○D8, C8⋊C22, C2×D12, C4○D12, C2×C3⋊D4, D4.2D4, C12⋊7D4, D12⋊6C22, Q8.13D6, D4.1D12
►On 96 points
(1 32 42 50)(2 33 43 51)(3 34 44 52)(4 35 45 53)(5 36 46 54)(6 25 47 55)(7 26 48 56)(8 27 37 57)(9 28 38 58)(10 29 39 59)(11 30 40 60)(12 31 41 49)(13 78 94 64)(14 79 95 65)(15 80 96 66)(16 81 85 67)(17 82 86 68)(18 83 87 69)(19 84 88 70)(20 73 89 71)(21 74 90 72)(22 75 91 61)(23 76 92 62)(24 77 93 63)
(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 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 94)(26 95)(27 96)(28 85)(29 86)(30 87)(31 88)(32 89)(33 90)(34 91)(35 92)(36 93)(37 80)(38 81)(39 82)(40 83)(41 84)(42 73)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)
(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 41 42 12)(2 11 43 40)(3 39 44 10)(4 9 45 38)(5 37 46 8)(6 7 47 48)(13 79 94 65)(14 64 95 78)(15 77 96 63)(16 62 85 76)(17 75 86 61)(18 72 87 74)(19 73 88 71)(20 70 89 84)(21 83 90 69)(22 68 91 82)(23 81 92 67)(24 66 93 80)(25 56 55 26)(27 54 57 36)(28 35 58 53)(29 52 59 34)(30 33 60 51)(31 50 49 32)
G:=sub<Sym(96)| (1,32,42,50)(2,33,43,51)(3,34,44,52)(4,35,45,53)(5,36,46,54)(6,25,47,55)(7,26,48,56)(8,27,37,57)(9,28,38,58)(10,29,39,59)(11,30,40,60)(12,31,41,49)(13,78,94,64)(14,79,95,65)(15,80,96,66)(16,81,85,67)(17,82,86,68)(18,83,87,69)(19,84,88,70)(20,73,89,71)(21,74,90,72)(22,75,91,61)(23,76,92,62)(24,77,93,63), (1,71)(2,72)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,94)(26,95)(27,96)(28,85)(29,86)(30,87)(31,88)(32,89)(33,90)(34,91)(35,92)(36,93)(37,80)(38,81)(39,82)(40,83)(41,84)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79), (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), (1,41,42,12)(2,11,43,40)(3,39,44,10)(4,9,45,38)(5,37,46,8)(6,7,47,48)(13,79,94,65)(14,64,95,78)(15,77,96,63)(16,62,85,76)(17,75,86,61)(18,72,87,74)(19,73,88,71)(20,70,89,84)(21,83,90,69)(22,68,91,82)(23,81,92,67)(24,66,93,80)(25,56,55,26)(27,54,57,36)(28,35,58,53)(29,52,59,34)(30,33,60,51)(31,50,49,32)>;
G:=Group( (1,32,42,50)(2,33,43,51)(3,34,44,52)(4,35,45,53)(5,36,46,54)(6,25,47,55)(7,26,48,56)(8,27,37,57)(9,28,38,58)(10,29,39,59)(11,30,40,60)(12,31,41,49)(13,78,94,64)(14,79,95,65)(15,80,96,66)(16,81,85,67)(17,82,86,68)(18,83,87,69)(19,84,88,70)(20,73,89,71)(21,74,90,72)(22,75,91,61)(23,76,92,62)(24,77,93,63), (1,71)(2,72)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,94)(26,95)(27,96)(28,85)(29,86)(30,87)(31,88)(32,89)(33,90)(34,91)(35,92)(36,93)(37,80)(38,81)(39,82)(40,83)(41,84)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79), (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), (1,41,42,12)(2,11,43,40)(3,39,44,10)(4,9,45,38)(5,37,46,8)(6,7,47,48)(13,79,94,65)(14,64,95,78)(15,77,96,63)(16,62,85,76)(17,75,86,61)(18,72,87,74)(19,73,88,71)(20,70,89,84)(21,83,90,69)(22,68,91,82)(23,81,92,67)(24,66,93,80)(25,56,55,26)(27,54,57,36)(28,35,58,53)(29,52,59,34)(30,33,60,51)(31,50,49,32) );
G=PermutationGroup([[(1,32,42,50),(2,33,43,51),(3,34,44,52),(4,35,45,53),(5,36,46,54),(6,25,47,55),(7,26,48,56),(8,27,37,57),(9,28,38,58),(10,29,39,59),(11,30,40,60),(12,31,41,49),(13,78,94,64),(14,79,95,65),(15,80,96,66),(16,81,85,67),(17,82,86,68),(18,83,87,69),(19,84,88,70),(20,73,89,71),(21,74,90,72),(22,75,91,61),(23,76,92,62),(24,77,93,63)], [(1,71),(2,72),(3,61),(4,62),(5,63),(6,64),(7,65),(8,66),(9,67),(10,68),(11,69),(12,70),(13,55),(14,56),(15,57),(16,58),(17,59),(18,60),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,94),(26,95),(27,96),(28,85),(29,86),(30,87),(31,88),(32,89),(33,90),(34,91),(35,92),(36,93),(37,80),(38,81),(39,82),(40,83),(41,84),(42,73),(43,74),(44,75),(45,76),(46,77),(47,78),(48,79)], [(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)], [(1,41,42,12),(2,11,43,40),(3,39,44,10),(4,9,45,38),(5,37,46,8),(6,7,47,48),(13,79,94,65),(14,64,95,78),(15,77,96,63),(16,62,85,76),(17,75,86,61),(18,72,87,74),(19,73,88,71),(20,70,89,84),(21,83,90,69),(22,68,91,82),(23,81,92,67),(24,66,93,80),(25,56,55,26),(27,54,57,36),(28,35,58,53),(29,52,59,34),(30,33,60,51),(31,50,49,32)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 24 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | D12 | C4○D8 | C4○D12 | C8⋊C22 | D12⋊6C22 | Q8.13D6 |
| kernel | D4.1D12 | C12⋊C8 | C6.D8 | C6.SD16 | C42⋊7S3 | C2×D4⋊S3 | C2×D4.S3 | D4×C12 | C4×D4 | C2×C12 | C3×D4 | C42 | C4⋊C4 | C2×D4 | C12 | C2×C4 | D4 | C6 | C4 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 1 | 2 | 2 |
Matrix representation of D4.1D12 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 71 |
| 0 | 0 | 1 | 1 |
| 30 | 13 | 0 | 0 |
| 60 | 43 | 0 | 0 |
| 0 | 0 | 32 | 32 |
| 0 | 0 | 57 | 41 |
| 59 | 7 | 0 | 0 |
| 66 | 66 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 66 | 66 | 0 | 0 |
| 59 | 7 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 27 | 27 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,1,0,0,71,1],[30,60,0,0,13,43,0,0,0,0,32,57,0,0,32,41],[59,66,0,0,7,66,0,0,0,0,46,0,0,0,0,46],[66,59,0,0,66,7,0,0,0,0,46,27,0,0,0,27] >;
D4.1D12 in GAP, Magma, Sage, TeX
D_4._1D_{12} % in TeX
G:=Group("D4.1D12"); // GroupNames label
G:=SmallGroup(192,575);
// by ID
G=gap.SmallGroup(192,575);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^12=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=a^2*c^-1>;
// generators/relations