metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.3Q8, C42.38D6, C4⋊C8⋊7S3, C3⋊3(D4.Q8), C4.45(S3×Q8), C24⋊1C4⋊13C2, C8⋊Dic3⋊17C2, (C2×C8).132D6, (C2×C4).40D12, (C4×D12).13C2, C6.14(C4○D8), (C2×C12).246D4, C12.104(C2×Q8), C2.D24.4C2, C2.16(C4○D24), C2.19(C8⋊D6), C6.16(C8⋊C22), (C4×C12).73C22, (C2×C24).26C22, C12.6Q8⋊10C2, C6.32(C22⋊Q8), C12.288(C4○D4), (C2×C12).757C23, C2.13(C4.D12), C22.120(C2×D12), C4⋊Dic3.20C22, C4.112(D4⋊2S3), (C2×D12).197C22, (C3×C4⋊C8)⋊9C2, (C2×C6).140(C2×D4), (C2×C4).702(C22×S3), SmallGroup(192,406)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.3Q8
G = < a,b,c,d | a12=b2=1, c4=a6, d2=a9c2, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=a3c3 >
Subgroups: 328 in 102 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C24, C4×S3, D12, D12, C2×Dic3, C2×C12, C22×S3, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C4×C12, C2×C24, S3×C2×C4, C2×D12, D4.Q8, C8⋊Dic3, C24⋊1C4, C2.D24, C3×C4⋊C8, C12.6Q8, C4×D12, D12.3Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, D12, C22×S3, C22⋊Q8, C4○D8, C8⋊C22, C2×D12, D4⋊2S3, S3×Q8, D4.Q8, C4.D12, C4○D24, C8⋊D6, D12.3Q8
►On 96 points
Generators in S96
(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 73)(2 84)(3 83)(4 82)(5 81)(6 80)(7 79)(8 78)(9 77)(10 76)(11 75)(12 74)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)(37 54)(38 53)(39 52)(40 51)(41 50)(42 49)(43 60)(44 59)(45 58)(46 57)(47 56)(48 55)(61 93)(62 92)(63 91)(64 90)(65 89)(66 88)(67 87)(68 86)(69 85)(70 96)(71 95)(72 94)
(1 13 77 25 7 19 83 31)(2 14 78 26 8 20 84 32)(3 15 79 27 9 21 73 33)(4 16 80 28 10 22 74 34)(5 17 81 29 11 23 75 35)(6 18 82 30 12 24 76 36)(37 90 53 66 43 96 59 72)(38 91 54 67 44 85 60 61)(39 92 55 68 45 86 49 62)(40 93 56 69 46 87 50 63)(41 94 57 70 47 88 51 64)(42 95 58 71 48 89 52 65)
(1 70 74 85)(2 71 75 86)(3 72 76 87)(4 61 77 88)(5 62 78 89)(6 63 79 90)(7 64 80 91)(8 65 81 92)(9 66 82 93)(10 67 83 94)(11 68 84 95)(12 69 73 96)(13 54 34 41)(14 55 35 42)(15 56 36 43)(16 57 25 44)(17 58 26 45)(18 59 27 46)(19 60 28 47)(20 49 29 48)(21 50 30 37)(22 51 31 38)(23 52 32 39)(24 53 33 40)
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), (1,73)(2,84)(3,83)(4,82)(5,81)(6,80)(7,79)(8,78)(9,77)(10,76)(11,75)(12,74)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(61,93)(62,92)(63,91)(64,90)(65,89)(66,88)(67,87)(68,86)(69,85)(70,96)(71,95)(72,94), (1,13,77,25,7,19,83,31)(2,14,78,26,8,20,84,32)(3,15,79,27,9,21,73,33)(4,16,80,28,10,22,74,34)(5,17,81,29,11,23,75,35)(6,18,82,30,12,24,76,36)(37,90,53,66,43,96,59,72)(38,91,54,67,44,85,60,61)(39,92,55,68,45,86,49,62)(40,93,56,69,46,87,50,63)(41,94,57,70,47,88,51,64)(42,95,58,71,48,89,52,65), (1,70,74,85)(2,71,75,86)(3,72,76,87)(4,61,77,88)(5,62,78,89)(6,63,79,90)(7,64,80,91)(8,65,81,92)(9,66,82,93)(10,67,83,94)(11,68,84,95)(12,69,73,96)(13,54,34,41)(14,55,35,42)(15,56,36,43)(16,57,25,44)(17,58,26,45)(18,59,27,46)(19,60,28,47)(20,49,29,48)(21,50,30,37)(22,51,31,38)(23,52,32,39)(24,53,33,40)>;
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), (1,73)(2,84)(3,83)(4,82)(5,81)(6,80)(7,79)(8,78)(9,77)(10,76)(11,75)(12,74)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(61,93)(62,92)(63,91)(64,90)(65,89)(66,88)(67,87)(68,86)(69,85)(70,96)(71,95)(72,94), (1,13,77,25,7,19,83,31)(2,14,78,26,8,20,84,32)(3,15,79,27,9,21,73,33)(4,16,80,28,10,22,74,34)(5,17,81,29,11,23,75,35)(6,18,82,30,12,24,76,36)(37,90,53,66,43,96,59,72)(38,91,54,67,44,85,60,61)(39,92,55,68,45,86,49,62)(40,93,56,69,46,87,50,63)(41,94,57,70,47,88,51,64)(42,95,58,71,48,89,52,65), (1,70,74,85)(2,71,75,86)(3,72,76,87)(4,61,77,88)(5,62,78,89)(6,63,79,90)(7,64,80,91)(8,65,81,92)(9,66,82,93)(10,67,83,94)(11,68,84,95)(12,69,73,96)(13,54,34,41)(14,55,35,42)(15,56,36,43)(16,57,25,44)(17,58,26,45)(18,59,27,46)(19,60,28,47)(20,49,29,48)(21,50,30,37)(22,51,31,38)(23,52,32,39)(24,53,33,40) );
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)], [(1,73),(2,84),(3,83),(4,82),(5,81),(6,80),(7,79),(8,78),(9,77),(10,76),(11,75),(12,74),(13,30),(14,29),(15,28),(16,27),(17,26),(18,25),(19,36),(20,35),(21,34),(22,33),(23,32),(24,31),(37,54),(38,53),(39,52),(40,51),(41,50),(42,49),(43,60),(44,59),(45,58),(46,57),(47,56),(48,55),(61,93),(62,92),(63,91),(64,90),(65,89),(66,88),(67,87),(68,86),(69,85),(70,96),(71,95),(72,94)], [(1,13,77,25,7,19,83,31),(2,14,78,26,8,20,84,32),(3,15,79,27,9,21,73,33),(4,16,80,28,10,22,74,34),(5,17,81,29,11,23,75,35),(6,18,82,30,12,24,76,36),(37,90,53,66,43,96,59,72),(38,91,54,67,44,85,60,61),(39,92,55,68,45,86,49,62),(40,93,56,69,46,87,50,63),(41,94,57,70,47,88,51,64),(42,95,58,71,48,89,52,65)], [(1,70,74,85),(2,71,75,86),(3,72,76,87),(4,61,77,88),(5,62,78,89),(6,63,79,90),(7,64,80,91),(8,65,81,92),(9,66,82,93),(10,67,83,94),(11,68,84,95),(12,69,73,96),(13,54,34,41),(14,55,35,42),(15,56,36,43),(16,57,25,44),(17,58,26,45),(18,59,27,46),(19,60,28,47),(20,49,29,48),(21,50,30,37),(22,51,31,38),(23,52,32,39),(24,53,33,40)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 12 | 12 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D4 | D6 | D6 | C4○D4 | D12 | C4○D8 | C4○D24 | C8⋊C22 | D4⋊2S3 | S3×Q8 | C8⋊D6 |
| kernel | D12.3Q8 | C8⋊Dic3 | C24⋊1C4 | C2.D24 | C3×C4⋊C8 | C12.6Q8 | C4×D12 | C4⋊C8 | D12 | C2×C12 | C42 | C2×C8 | C12 | C2×C4 | C6 | C2 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 8 | 1 | 1 | 1 | 2 |
Matrix representation of D12.3Q8 ►in GL4(𝔽73) generated by
| 59 | 7 | 0 | 0 |
| 66 | 66 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 7 | 7 | 0 | 0 |
| 14 | 66 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 48 | 72 |
| 36 | 62 | 0 | 0 |
| 11 | 25 | 0 | 0 |
| 0 | 0 | 72 | 70 |
| 0 | 0 | 0 | 1 |
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 55 | 46 |
G:=sub<GL(4,GF(73))| [59,66,0,0,7,66,0,0,0,0,72,0,0,0,0,72],[7,14,0,0,7,66,0,0,0,0,1,48,0,0,0,72],[36,11,0,0,62,25,0,0,0,0,72,0,0,0,70,1],[46,0,0,0,0,46,0,0,0,0,27,55,0,0,0,46] >;
D12.3Q8 in GAP, Magma, Sage, TeX
D_{12}._3Q_8 % in TeX
G:=Group("D12.3Q8"); // GroupNames label
G:=SmallGroup(192,406);
// by ID
G=gap.SmallGroup(192,406);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,344,254,219,142,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=1,c^4=a^6,d^2=a^9*c^2,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^3*c^3>;
// generators/relations