metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.9D4, C4.89(S3×D4), C4⋊C4.141D6, (C2×D4).32D6, (C2×C8).118D6, D4⋊C4⋊14S3, Dic3⋊5D4⋊4C2, Dic3⋊C8⋊12C2, C6.43(C4○D8), C4.4(C4○D12), C6.SD16⋊8C2, C12.113(C2×D4), C3⋊1(D4.2D4), C12.12(C4○D4), C23.12D6⋊2C2, C6.20(C4⋊D4), C2.19(D8⋊S3), C6.37(C8⋊C22), (C2×Dic3).25D4, (C6×D4).48C22, C22.184(S3×D4), C2.23(Dic3⋊D4), (C2×C24).129C22, (C2×C12).227C23, (C2×D12).55C22, C2.13(Q8.7D6), (C4×Dic3).14C22, (C2×Dic6).61C22, (C2×D4⋊S3).3C2, (C2×C24⋊C2)⋊16C2, (C2×C6).240(C2×D4), (C2×C3⋊C8).25C22, (C3×D4⋊C4)⋊14C2, (C3×C4⋊C4).28C22, (C2×C4).334(C22×S3), SmallGroup(192,346)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.D4
G = < a,b,c,d | a12=b2=c4=1, d2=a9, bab=a-1, cac-1=a7, ad=da, bc=cb, dbd-1=a3b, dcd-1=a9c-1 >
Subgroups: 408 in 124 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C3⋊C8, C24, Dic6, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C24⋊C2, C2×C3⋊C8, C4×Dic3, D6⋊C4, D4⋊S3, C6.D4, C3×C4⋊C4, C2×C24, C2×Dic6, S3×C2×C4, C2×D12, C6×D4, D4.2D4, C6.SD16, Dic3⋊C8, C3×D4⋊C4, Dic3⋊5D4, C2×C24⋊C2, C2×D4⋊S3, C23.12D6, D12.D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C4⋊D4, C4○D8, C8⋊C22, C4○D12, S3×D4, D4.2D4, Dic3⋊D4, D8⋊S3, Q8.7D6, D12.D4
►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 81)(2 80)(3 79)(4 78)(5 77)(6 76)(7 75)(8 74)(9 73)(10 84)(11 83)(12 82)(13 94)(14 93)(15 92)(16 91)(17 90)(18 89)(19 88)(20 87)(21 86)(22 85)(23 96)(24 95)(25 37)(26 48)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)(33 41)(34 40)(35 39)(36 38)(49 65)(50 64)(51 63)(52 62)(53 61)(54 72)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)
(1 23 73 88)(2 18 74 95)(3 13 75 90)(4 20 76 85)(5 15 77 92)(6 22 78 87)(7 17 79 94)(8 24 80 89)(9 19 81 96)(10 14 82 91)(11 21 83 86)(12 16 84 93)(25 56 38 65)(26 51 39 72)(27 58 40 67)(28 53 41 62)(29 60 42 69)(30 55 43 64)(31 50 44 71)(32 57 45 66)(33 52 46 61)(34 59 47 68)(35 54 48 63)(36 49 37 70)
(1 34 10 31 7 28 4 25)(2 35 11 32 8 29 5 26)(3 36 12 33 9 30 6 27)(13 67 22 64 19 61 16 70)(14 68 23 65 20 62 17 71)(15 69 24 66 21 63 18 72)(37 84 46 81 43 78 40 75)(38 73 47 82 44 79 41 76)(39 74 48 83 45 80 42 77)(49 90 58 87 55 96 52 93)(50 91 59 88 56 85 53 94)(51 92 60 89 57 86 54 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), (1,81)(2,80)(3,79)(4,78)(5,77)(6,76)(7,75)(8,74)(9,73)(10,84)(11,83)(12,82)(13,94)(14,93)(15,92)(16,91)(17,90)(18,89)(19,88)(20,87)(21,86)(22,85)(23,96)(24,95)(25,37)(26,48)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(49,65)(50,64)(51,63)(52,62)(53,61)(54,72)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66), (1,23,73,88)(2,18,74,95)(3,13,75,90)(4,20,76,85)(5,15,77,92)(6,22,78,87)(7,17,79,94)(8,24,80,89)(9,19,81,96)(10,14,82,91)(11,21,83,86)(12,16,84,93)(25,56,38,65)(26,51,39,72)(27,58,40,67)(28,53,41,62)(29,60,42,69)(30,55,43,64)(31,50,44,71)(32,57,45,66)(33,52,46,61)(34,59,47,68)(35,54,48,63)(36,49,37,70), (1,34,10,31,7,28,4,25)(2,35,11,32,8,29,5,26)(3,36,12,33,9,30,6,27)(13,67,22,64,19,61,16,70)(14,68,23,65,20,62,17,71)(15,69,24,66,21,63,18,72)(37,84,46,81,43,78,40,75)(38,73,47,82,44,79,41,76)(39,74,48,83,45,80,42,77)(49,90,58,87,55,96,52,93)(50,91,59,88,56,85,53,94)(51,92,60,89,57,86,54,95)>;
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), (1,81)(2,80)(3,79)(4,78)(5,77)(6,76)(7,75)(8,74)(9,73)(10,84)(11,83)(12,82)(13,94)(14,93)(15,92)(16,91)(17,90)(18,89)(19,88)(20,87)(21,86)(22,85)(23,96)(24,95)(25,37)(26,48)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(49,65)(50,64)(51,63)(52,62)(53,61)(54,72)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66), (1,23,73,88)(2,18,74,95)(3,13,75,90)(4,20,76,85)(5,15,77,92)(6,22,78,87)(7,17,79,94)(8,24,80,89)(9,19,81,96)(10,14,82,91)(11,21,83,86)(12,16,84,93)(25,56,38,65)(26,51,39,72)(27,58,40,67)(28,53,41,62)(29,60,42,69)(30,55,43,64)(31,50,44,71)(32,57,45,66)(33,52,46,61)(34,59,47,68)(35,54,48,63)(36,49,37,70), (1,34,10,31,7,28,4,25)(2,35,11,32,8,29,5,26)(3,36,12,33,9,30,6,27)(13,67,22,64,19,61,16,70)(14,68,23,65,20,62,17,71)(15,69,24,66,21,63,18,72)(37,84,46,81,43,78,40,75)(38,73,47,82,44,79,41,76)(39,74,48,83,45,80,42,77)(49,90,58,87,55,96,52,93)(50,91,59,88,56,85,53,94)(51,92,60,89,57,86,54,95) );
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)], [(1,81),(2,80),(3,79),(4,78),(5,77),(6,76),(7,75),(8,74),(9,73),(10,84),(11,83),(12,82),(13,94),(14,93),(15,92),(16,91),(17,90),(18,89),(19,88),(20,87),(21,86),(22,85),(23,96),(24,95),(25,37),(26,48),(27,47),(28,46),(29,45),(30,44),(31,43),(32,42),(33,41),(34,40),(35,39),(36,38),(49,65),(50,64),(51,63),(52,62),(53,61),(54,72),(55,71),(56,70),(57,69),(58,68),(59,67),(60,66)], [(1,23,73,88),(2,18,74,95),(3,13,75,90),(4,20,76,85),(5,15,77,92),(6,22,78,87),(7,17,79,94),(8,24,80,89),(9,19,81,96),(10,14,82,91),(11,21,83,86),(12,16,84,93),(25,56,38,65),(26,51,39,72),(27,58,40,67),(28,53,41,62),(29,60,42,69),(30,55,43,64),(31,50,44,71),(32,57,45,66),(33,52,46,61),(34,59,47,68),(35,54,48,63),(36,49,37,70)], [(1,34,10,31,7,28,4,25),(2,35,11,32,8,29,5,26),(3,36,12,33,9,30,6,27),(13,67,22,64,19,61,16,70),(14,68,23,65,20,62,17,71),(15,69,24,66,21,63,18,72),(37,84,46,81,43,78,40,75),(38,73,47,82,44,79,41,76),(39,74,48,83,45,80,42,77),(49,90,58,87,55,96,52,93),(50,91,59,88,56,85,53,94),(51,92,60,89,57,86,54,95)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 8 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 24 | 2 | 2 | 2 | 8 | 8 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | C4○D4 | C4○D8 | C4○D12 | C8⋊C22 | S3×D4 | S3×D4 | D8⋊S3 | Q8.7D6 |
| kernel | D12.D4 | C6.SD16 | Dic3⋊C8 | C3×D4⋊C4 | Dic3⋊5D4 | C2×C24⋊C2 | C2×D4⋊S3 | C23.12D6 | D4⋊C4 | D12 | C2×Dic3 | C4⋊C4 | C2×C8 | C2×D4 | C12 | C6 | C4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of D12.D4 ►in GL4(𝔽73) generated by
| 72 | 70 | 0 | 0 |
| 25 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 72 | 0 |
| 72 | 0 | 0 | 0 |
| 25 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 72 |
| 27 | 0 | 0 | 0 |
| 55 | 46 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
| 12 | 18 | 0 | 0 |
| 69 | 0 | 0 | 0 |
| 0 | 0 | 7 | 14 |
| 0 | 0 | 59 | 66 |
G:=sub<GL(4,GF(73))| [72,25,0,0,70,1,0,0,0,0,1,72,0,0,1,0],[72,25,0,0,0,1,0,0,0,0,1,0,0,0,1,72],[27,55,0,0,0,46,0,0,0,0,27,0,0,0,0,27],[12,69,0,0,18,0,0,0,0,0,7,59,0,0,14,66] >;
D12.D4 in GAP, Magma, Sage, TeX
D_{12}.D_4 % in TeX
G:=Group("D12.D4"); // GroupNames label
G:=SmallGroup(192,346);
// by ID
G=gap.SmallGroup(192,346);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,590,555,1684,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=1,d^2=a^9,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=a^9*c^-1>;
// generators/relations