metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.32D4, Dic6.31D4, C23.16D12, C22⋊C8⋊5S3, C6.9C22≀C2, C6.8(C4○D8), C2.D24⋊5C2, (C2×C8).108D6, C4.121(S3×D4), (C2×Dic12)⋊2C2, (C2×C4).118D12, (C2×C12).240D4, C12.333(C2×D4), C3⋊1(D4.7D4), (C2×C24).3C22, (C22×C4).99D6, (C22×C6).53D4, C2.10(C4○D24), C2.Dic12⋊10C2, C6.9(C8.C22), C12.48D4⋊16C2, C2.12(D6⋊D4), (C2×C12).743C23, C2.12(C8.D6), C22.106(C2×D12), C4⋊Dic3.12C22, (C2×D12).192C22, (C22×C12).96C22, (C2×Dic6).210C22, (C3×C22⋊C8)⋊7C2, (C2×C24⋊C2)⋊10C2, (C2×C4○D12).2C2, (C2×C6).126(C2×D4), (C2×C4).688(C22×S3), SmallGroup(192,292)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.32D4
G = < a,b,c,d | a12=b2=1, c4=d2=a6, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=a9c3 >
Subgroups: 480 in 152 conjugacy classes, 43 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C24⋊C2, Dic12, Dic3⋊C4, C4⋊Dic3, C6.D4, C2×C24, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, D4.7D4, C2.Dic12, C2.D24, C3×C22⋊C8, C2×C24⋊C2, C2×Dic12, C12.48D4, C2×C4○D12, D12.32D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C22≀C2, C4○D8, C8.C22, C2×D12, S3×D4, D4.7D4, D6⋊D4, C4○D24, C8.D6, D12.32D4
►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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 18)(14 17)(15 16)(19 24)(20 23)(21 22)(25 30)(26 29)(27 28)(31 36)(32 35)(33 34)(37 46)(38 45)(39 44)(40 43)(41 42)(47 48)(49 57)(50 56)(51 55)(52 54)(58 60)(61 63)(64 72)(65 71)(66 70)(67 69)(73 79)(74 78)(75 77)(80 84)(81 83)(85 91)(86 90)(87 89)(92 96)(93 95)
(1 64 45 81 7 70 39 75)(2 65 46 82 8 71 40 76)(3 66 47 83 9 72 41 77)(4 67 48 84 10 61 42 78)(5 68 37 73 11 62 43 79)(6 69 38 74 12 63 44 80)(13 52 28 90 19 58 34 96)(14 53 29 91 20 59 35 85)(15 54 30 92 21 60 36 86)(16 55 31 93 22 49 25 87)(17 56 32 94 23 50 26 88)(18 57 33 95 24 51 27 89)
(1 28 7 34)(2 29 8 35)(3 30 9 36)(4 31 10 25)(5 32 11 26)(6 33 12 27)(13 45 19 39)(14 46 20 40)(15 47 21 41)(16 48 22 42)(17 37 23 43)(18 38 24 44)(49 64 55 70)(50 65 56 71)(51 66 57 72)(52 67 58 61)(53 68 59 62)(54 69 60 63)(73 85 79 91)(74 86 80 92)(75 87 81 93)(76 88 82 94)(77 89 83 95)(78 90 84 96)
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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)(25,30)(26,29)(27,28)(31,36)(32,35)(33,34)(37,46)(38,45)(39,44)(40,43)(41,42)(47,48)(49,57)(50,56)(51,55)(52,54)(58,60)(61,63)(64,72)(65,71)(66,70)(67,69)(73,79)(74,78)(75,77)(80,84)(81,83)(85,91)(86,90)(87,89)(92,96)(93,95), (1,64,45,81,7,70,39,75)(2,65,46,82,8,71,40,76)(3,66,47,83,9,72,41,77)(4,67,48,84,10,61,42,78)(5,68,37,73,11,62,43,79)(6,69,38,74,12,63,44,80)(13,52,28,90,19,58,34,96)(14,53,29,91,20,59,35,85)(15,54,30,92,21,60,36,86)(16,55,31,93,22,49,25,87)(17,56,32,94,23,50,26,88)(18,57,33,95,24,51,27,89), (1,28,7,34)(2,29,8,35)(3,30,9,36)(4,31,10,25)(5,32,11,26)(6,33,12,27)(13,45,19,39)(14,46,20,40)(15,47,21,41)(16,48,22,42)(17,37,23,43)(18,38,24,44)(49,64,55,70)(50,65,56,71)(51,66,57,72)(52,67,58,61)(53,68,59,62)(54,69,60,63)(73,85,79,91)(74,86,80,92)(75,87,81,93)(76,88,82,94)(77,89,83,95)(78,90,84,96)>;
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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)(25,30)(26,29)(27,28)(31,36)(32,35)(33,34)(37,46)(38,45)(39,44)(40,43)(41,42)(47,48)(49,57)(50,56)(51,55)(52,54)(58,60)(61,63)(64,72)(65,71)(66,70)(67,69)(73,79)(74,78)(75,77)(80,84)(81,83)(85,91)(86,90)(87,89)(92,96)(93,95), (1,64,45,81,7,70,39,75)(2,65,46,82,8,71,40,76)(3,66,47,83,9,72,41,77)(4,67,48,84,10,61,42,78)(5,68,37,73,11,62,43,79)(6,69,38,74,12,63,44,80)(13,52,28,90,19,58,34,96)(14,53,29,91,20,59,35,85)(15,54,30,92,21,60,36,86)(16,55,31,93,22,49,25,87)(17,56,32,94,23,50,26,88)(18,57,33,95,24,51,27,89), (1,28,7,34)(2,29,8,35)(3,30,9,36)(4,31,10,25)(5,32,11,26)(6,33,12,27)(13,45,19,39)(14,46,20,40)(15,47,21,41)(16,48,22,42)(17,37,23,43)(18,38,24,44)(49,64,55,70)(50,65,56,71)(51,66,57,72)(52,67,58,61)(53,68,59,62)(54,69,60,63)(73,85,79,91)(74,86,80,92)(75,87,81,93)(76,88,82,94)(77,89,83,95)(78,90,84,96) );
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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,18),(14,17),(15,16),(19,24),(20,23),(21,22),(25,30),(26,29),(27,28),(31,36),(32,35),(33,34),(37,46),(38,45),(39,44),(40,43),(41,42),(47,48),(49,57),(50,56),(51,55),(52,54),(58,60),(61,63),(64,72),(65,71),(66,70),(67,69),(73,79),(74,78),(75,77),(80,84),(81,83),(85,91),(86,90),(87,89),(92,96),(93,95)], [(1,64,45,81,7,70,39,75),(2,65,46,82,8,71,40,76),(3,66,47,83,9,72,41,77),(4,67,48,84,10,61,42,78),(5,68,37,73,11,62,43,79),(6,69,38,74,12,63,44,80),(13,52,28,90,19,58,34,96),(14,53,29,91,20,59,35,85),(15,54,30,92,21,60,36,86),(16,55,31,93,22,49,25,87),(17,56,32,94,23,50,26,88),(18,57,33,95,24,51,27,89)], [(1,28,7,34),(2,29,8,35),(3,30,9,36),(4,31,10,25),(5,32,11,26),(6,33,12,27),(13,45,19,39),(14,46,20,40),(15,47,21,41),(16,48,22,42),(17,37,23,43),(18,38,24,44),(49,64,55,70),(50,65,56,71),(51,66,57,72),(52,67,58,61),(53,68,59,62),(54,69,60,63),(73,85,79,91),(74,86,80,92),(75,87,81,93),(76,88,82,94),(77,89,83,95),(78,90,84,96)]])
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 | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| 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 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | D4 | D4 | D6 | D6 | D12 | D12 | C4○D8 | C4○D24 | C8.C22 | S3×D4 | C8.D6 |
| kernel | D12.32D4 | C2.Dic12 | C2.D24 | C3×C22⋊C8 | C2×C24⋊C2 | C2×Dic12 | C12.48D4 | C2×C4○D12 | C22⋊C8 | Dic6 | D12 | C2×C12 | C22×C6 | C2×C8 | C22×C4 | C2×C4 | C23 | C6 | C2 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 8 | 1 | 2 | 2 |
Matrix representation of D12.32D4 ►in GL4(𝔽73) generated by
| 66 | 7 | 0 | 0 |
| 66 | 59 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 66 | 7 | 0 | 0 |
| 14 | 7 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 60 | 1 |
| 55 | 5 | 0 | 0 |
| 68 | 50 | 0 | 0 |
| 0 | 0 | 13 | 71 |
| 0 | 0 | 11 | 60 |
| 27 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 60 | 1 |
G:=sub<GL(4,GF(73))| [66,66,0,0,7,59,0,0,0,0,72,0,0,0,0,72],[66,14,0,0,7,7,0,0,0,0,72,60,0,0,0,1],[55,68,0,0,5,50,0,0,0,0,13,11,0,0,71,60],[27,0,0,0,0,27,0,0,0,0,72,60,0,0,0,1] >;
D12.32D4 in GAP, Magma, Sage, TeX
D_{12}._{32}D_4 % in TeX
G:=Group("D12.32D4"); // GroupNames label
G:=SmallGroup(192,292);
// by ID
G=gap.SmallGroup(192,292);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,58,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=1,c^4=d^2=a^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=a^9*c^3>;
// generators/relations