metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.2Q8, C2.D8⋊9S3, C4.8(S3×Q8), (C2×C8).32D6, C3⋊5(D4.Q8), C4⋊C4.174D6, C12.24(C2×Q8), Dic3⋊C8⋊16C2, C6.77(C4○D8), C4.Dic6⋊7C2, C6.D8.9C2, C2.D24.7C2, Dic3⋊5D4.9C2, C4.85(C4○D12), C2.25(D8⋊S3), C6.45(C8⋊C22), C12.Q8⋊23C2, (C2×Dic3).52D4, C22.236(S3×D4), C6.42(C22⋊Q8), C12.173(C4○D4), (C2×C24).174C22, (C2×C12).307C23, C2.19(D6⋊Q8), (C2×D12).87C22, C2.15(D24⋊C2), C4⋊Dic3.129C22, (C4×Dic3).39C22, (C3×C2.D8)⋊16C2, (C2×C6).312(C2×D4), (C2×C3⋊C8).76C22, (C3×C4⋊C4).100C22, (C2×C4).410(C22×S3), SmallGroup(192,450)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.2Q8
G = < a,b,c,d | a12=b2=1, c4=a6, d2=a3c2, bab=a-1, ac=ca, dad-1=a7, cbc-1=a3b, bd=db, dcd-1=a6c3 >
Subgroups: 320 in 102 conjugacy classes, 39 normal (37 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, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C3⋊C8, C24, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, S3×C2×C4, C2×D12, D4.Q8, C12.Q8, C6.D8, Dic3⋊C8, C2.D24, C3×C2.D8, C4.Dic6, Dic3⋊5D4, D12.2Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, C22×S3, C22⋊Q8, C4○D8, C8⋊C22, C4○D12, S3×D4, S3×Q8, D4.Q8, D6⋊Q8, D8⋊S3, D24⋊C2, D12.2Q8
►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 15)(2 14)(3 13)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)(25 84)(26 83)(27 82)(28 81)(29 80)(30 79)(31 78)(32 77)(33 76)(34 75)(35 74)(36 73)(37 52)(38 51)(39 50)(40 49)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(61 85)(62 96)(63 95)(64 94)(65 93)(66 92)(67 91)(68 90)(69 89)(70 88)(71 87)(72 86)
(1 25 16 76 7 31 22 82)(2 26 17 77 8 32 23 83)(3 27 18 78 9 33 24 84)(4 28 19 79 10 34 13 73)(5 29 20 80 11 35 14 74)(6 30 21 81 12 36 15 75)(37 62 49 96 43 68 55 90)(38 63 50 85 44 69 56 91)(39 64 51 86 45 70 57 92)(40 65 52 87 46 71 58 93)(41 66 53 88 47 72 59 94)(42 67 54 89 48 61 60 95)
(1 39 19 54)(2 46 20 49)(3 41 21 56)(4 48 22 51)(5 43 23 58)(6 38 24 53)(7 45 13 60)(8 40 14 55)(9 47 15 50)(10 42 16 57)(11 37 17 52)(12 44 18 59)(25 92 79 67)(26 87 80 62)(27 94 81 69)(28 89 82 64)(29 96 83 71)(30 91 84 66)(31 86 73 61)(32 93 74 68)(33 88 75 63)(34 95 76 70)(35 90 77 65)(36 85 78 72)
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), (1,15)(2,14)(3,13)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(25,84)(26,83)(27,82)(28,81)(29,80)(30,79)(31,78)(32,77)(33,76)(34,75)(35,74)(36,73)(37,52)(38,51)(39,50)(40,49)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(61,85)(62,96)(63,95)(64,94)(65,93)(66,92)(67,91)(68,90)(69,89)(70,88)(71,87)(72,86), (1,25,16,76,7,31,22,82)(2,26,17,77,8,32,23,83)(3,27,18,78,9,33,24,84)(4,28,19,79,10,34,13,73)(5,29,20,80,11,35,14,74)(6,30,21,81,12,36,15,75)(37,62,49,96,43,68,55,90)(38,63,50,85,44,69,56,91)(39,64,51,86,45,70,57,92)(40,65,52,87,46,71,58,93)(41,66,53,88,47,72,59,94)(42,67,54,89,48,61,60,95), (1,39,19,54)(2,46,20,49)(3,41,21,56)(4,48,22,51)(5,43,23,58)(6,38,24,53)(7,45,13,60)(8,40,14,55)(9,47,15,50)(10,42,16,57)(11,37,17,52)(12,44,18,59)(25,92,79,67)(26,87,80,62)(27,94,81,69)(28,89,82,64)(29,96,83,71)(30,91,84,66)(31,86,73,61)(32,93,74,68)(33,88,75,63)(34,95,76,70)(35,90,77,65)(36,85,78,72)>;
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), (1,15)(2,14)(3,13)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(25,84)(26,83)(27,82)(28,81)(29,80)(30,79)(31,78)(32,77)(33,76)(34,75)(35,74)(36,73)(37,52)(38,51)(39,50)(40,49)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(61,85)(62,96)(63,95)(64,94)(65,93)(66,92)(67,91)(68,90)(69,89)(70,88)(71,87)(72,86), (1,25,16,76,7,31,22,82)(2,26,17,77,8,32,23,83)(3,27,18,78,9,33,24,84)(4,28,19,79,10,34,13,73)(5,29,20,80,11,35,14,74)(6,30,21,81,12,36,15,75)(37,62,49,96,43,68,55,90)(38,63,50,85,44,69,56,91)(39,64,51,86,45,70,57,92)(40,65,52,87,46,71,58,93)(41,66,53,88,47,72,59,94)(42,67,54,89,48,61,60,95), (1,39,19,54)(2,46,20,49)(3,41,21,56)(4,48,22,51)(5,43,23,58)(6,38,24,53)(7,45,13,60)(8,40,14,55)(9,47,15,50)(10,42,16,57)(11,37,17,52)(12,44,18,59)(25,92,79,67)(26,87,80,62)(27,94,81,69)(28,89,82,64)(29,96,83,71)(30,91,84,66)(31,86,73,61)(32,93,74,68)(33,88,75,63)(34,95,76,70)(35,90,77,65)(36,85,78,72) );
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)], [(1,15),(2,14),(3,13),(4,24),(5,23),(6,22),(7,21),(8,20),(9,19),(10,18),(11,17),(12,16),(25,84),(26,83),(27,82),(28,81),(29,80),(30,79),(31,78),(32,77),(33,76),(34,75),(35,74),(36,73),(37,52),(38,51),(39,50),(40,49),(41,60),(42,59),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53),(61,85),(62,96),(63,95),(64,94),(65,93),(66,92),(67,91),(68,90),(69,89),(70,88),(71,87),(72,86)], [(1,25,16,76,7,31,22,82),(2,26,17,77,8,32,23,83),(3,27,18,78,9,33,24,84),(4,28,19,79,10,34,13,73),(5,29,20,80,11,35,14,74),(6,30,21,81,12,36,15,75),(37,62,49,96,43,68,55,90),(38,63,50,85,44,69,56,91),(39,64,51,86,45,70,57,92),(40,65,52,87,46,71,58,93),(41,66,53,88,47,72,59,94),(42,67,54,89,48,61,60,95)], [(1,39,19,54),(2,46,20,49),(3,41,21,56),(4,48,22,51),(5,43,23,58),(6,38,24,53),(7,45,13,60),(8,40,14,55),(9,47,15,50),(10,42,16,57),(11,37,17,52),(12,44,18,59),(25,92,79,67),(26,87,80,62),(27,94,81,69),(28,89,82,64),(29,96,83,71),(30,91,84,66),(31,86,73,61),(32,93,74,68),(33,88,75,63),(34,95,76,70),(35,90,77,65),(36,85,78,72)]])
33 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 | 24A | 24B | 24C | 24D |
| 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 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 8 | 12 | 24 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 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 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D4 | D6 | D6 | C4○D4 | C4○D8 | C4○D12 | C8⋊C22 | S3×Q8 | S3×D4 | D8⋊S3 | D24⋊C2 |
| kernel | D12.2Q8 | C12.Q8 | C6.D8 | Dic3⋊C8 | C2.D24 | C3×C2.D8 | C4.Dic6 | Dic3⋊5D4 | C2.D8 | D12 | C2×Dic3 | C4⋊C4 | C2×C8 | C12 | C6 | C4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 2 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of D12.2Q8 ►in GL4(𝔽73) generated by
| 0 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 72 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 72 |
| 16 | 16 | 0 | 0 |
| 57 | 16 | 0 | 0 |
| 0 | 0 | 30 | 60 |
| 0 | 0 | 13 | 43 |
| 27 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
G:=sub<GL(4,GF(73))| [0,1,0,0,72,0,0,0,0,0,1,72,0,0,1,0],[72,0,0,0,0,1,0,0,0,0,1,0,0,0,1,72],[16,57,0,0,16,16,0,0,0,0,30,13,0,0,60,43],[27,0,0,0,0,46,0,0,0,0,46,0,0,0,0,46] >;
D12.2Q8 in GAP, Magma, Sage, TeX
D_{12}._2Q_8 % in TeX
G:=Group("D12.2Q8"); // GroupNames label
G:=SmallGroup(192,450);
// by ID
G=gap.SmallGroup(192,450);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,590,555,100,1684,851,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=1,c^4=a^6,d^2=a^3*c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=a^6*c^3>;
// generators/relations