metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.10D4, Q8.11D12, D6⋊C8⋊14C2, C4⋊C4.25D6, C4.93(S3×D4), C4.6(C2×D12), (C3×Q8).1D4, C4.D12⋊5C2, (C2×C8).124D6, Q8⋊C4⋊12S3, C6.24C22≀C2, C6.48(C4○D8), C12.122(C2×D4), C6.D8⋊11C2, C3⋊3(D4.7D4), (C2×Q8).132D6, (C22×S3).17D4, C22.198(S3×D4), (C6×Q8).31C22, C2.27(D6⋊D4), (C2×C24).135C22, (C2×C12).248C23, (C2×Dic3).154D4, C2.15(Q16⋊S3), (C2×D12).64C22, C6.61(C8.C22), C2.17(Q8.7D6), (C2×Dic6).71C22, (C2×C3⋊Q16)⋊3C2, (C2×C24⋊C2)⋊18C2, (C2×C6).261(C2×D4), (C2×C3⋊C8).39C22, (S3×C2×C4).21C22, (C3×Q8⋊C4)⋊12C2, (C3×C4⋊C4).49C22, (C2×Q8⋊3S3).4C2, (C2×C4).355(C22×S3), SmallGroup(192,367)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.11D12
G = < a,b,c,d | a4=c12=1, b2=d2=a2, bab-1=cac-1=dad-1=a-1, cbc-1=a-1b, dbd-1=ab, dcd-1=a2c-1 >
Subgroups: 488 in 152 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×S3, C22×S3, C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C24⋊C2, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, C3⋊Q16, C3×C4⋊C4, C2×C24, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, Q8⋊3S3, C6×Q8, D4.7D4, C6.D8, D6⋊C8, C3×Q8⋊C4, C4.D12, C2×C24⋊C2, C2×C3⋊Q16, C2×Q8⋊3S3, Q8.11D12
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, Q8.7D6, Q16⋊S3, Q8.11D12
►On 96 points
(1 90 17 49)(2 50 18 91)(3 92 19 51)(4 52 20 93)(5 94 21 53)(6 54 22 95)(7 96 23 55)(8 56 24 85)(9 86 13 57)(10 58 14 87)(11 88 15 59)(12 60 16 89)(25 40 61 77)(26 78 62 41)(27 42 63 79)(28 80 64 43)(29 44 65 81)(30 82 66 45)(31 46 67 83)(32 84 68 47)(33 48 69 73)(34 74 70 37)(35 38 71 75)(36 76 72 39)
(1 39 17 76)(2 25 18 61)(3 41 19 78)(4 27 20 63)(5 43 21 80)(6 29 22 65)(7 45 23 82)(8 31 24 67)(9 47 13 84)(10 33 14 69)(11 37 15 74)(12 35 16 71)(26 92 62 51)(28 94 64 53)(30 96 66 55)(32 86 68 57)(34 88 70 59)(36 90 72 49)(38 60 75 89)(40 50 77 91)(42 52 79 93)(44 54 81 95)(46 56 83 85)(48 58 73 87)
(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 17 44)(2 43 18 80)(3 79 19 42)(4 41 20 78)(5 77 21 40)(6 39 22 76)(7 75 23 38)(8 37 24 74)(9 73 13 48)(10 47 14 84)(11 83 15 46)(12 45 16 82)(25 94 61 53)(26 52 62 93)(27 92 63 51)(28 50 64 91)(29 90 65 49)(30 60 66 89)(31 88 67 59)(32 58 68 87)(33 86 69 57)(34 56 70 85)(35 96 71 55)(36 54 72 95)
G:=sub<Sym(96)| (1,90,17,49)(2,50,18,91)(3,92,19,51)(4,52,20,93)(5,94,21,53)(6,54,22,95)(7,96,23,55)(8,56,24,85)(9,86,13,57)(10,58,14,87)(11,88,15,59)(12,60,16,89)(25,40,61,77)(26,78,62,41)(27,42,63,79)(28,80,64,43)(29,44,65,81)(30,82,66,45)(31,46,67,83)(32,84,68,47)(33,48,69,73)(34,74,70,37)(35,38,71,75)(36,76,72,39), (1,39,17,76)(2,25,18,61)(3,41,19,78)(4,27,20,63)(5,43,21,80)(6,29,22,65)(7,45,23,82)(8,31,24,67)(9,47,13,84)(10,33,14,69)(11,37,15,74)(12,35,16,71)(26,92,62,51)(28,94,64,53)(30,96,66,55)(32,86,68,57)(34,88,70,59)(36,90,72,49)(38,60,75,89)(40,50,77,91)(42,52,79,93)(44,54,81,95)(46,56,83,85)(48,58,73,87), (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), (1,81,17,44)(2,43,18,80)(3,79,19,42)(4,41,20,78)(5,77,21,40)(6,39,22,76)(7,75,23,38)(8,37,24,74)(9,73,13,48)(10,47,14,84)(11,83,15,46)(12,45,16,82)(25,94,61,53)(26,52,62,93)(27,92,63,51)(28,50,64,91)(29,90,65,49)(30,60,66,89)(31,88,67,59)(32,58,68,87)(33,86,69,57)(34,56,70,85)(35,96,71,55)(36,54,72,95)>;
G:=Group( (1,90,17,49)(2,50,18,91)(3,92,19,51)(4,52,20,93)(5,94,21,53)(6,54,22,95)(7,96,23,55)(8,56,24,85)(9,86,13,57)(10,58,14,87)(11,88,15,59)(12,60,16,89)(25,40,61,77)(26,78,62,41)(27,42,63,79)(28,80,64,43)(29,44,65,81)(30,82,66,45)(31,46,67,83)(32,84,68,47)(33,48,69,73)(34,74,70,37)(35,38,71,75)(36,76,72,39), (1,39,17,76)(2,25,18,61)(3,41,19,78)(4,27,20,63)(5,43,21,80)(6,29,22,65)(7,45,23,82)(8,31,24,67)(9,47,13,84)(10,33,14,69)(11,37,15,74)(12,35,16,71)(26,92,62,51)(28,94,64,53)(30,96,66,55)(32,86,68,57)(34,88,70,59)(36,90,72,49)(38,60,75,89)(40,50,77,91)(42,52,79,93)(44,54,81,95)(46,56,83,85)(48,58,73,87), (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), (1,81,17,44)(2,43,18,80)(3,79,19,42)(4,41,20,78)(5,77,21,40)(6,39,22,76)(7,75,23,38)(8,37,24,74)(9,73,13,48)(10,47,14,84)(11,83,15,46)(12,45,16,82)(25,94,61,53)(26,52,62,93)(27,92,63,51)(28,50,64,91)(29,90,65,49)(30,60,66,89)(31,88,67,59)(32,58,68,87)(33,86,69,57)(34,56,70,85)(35,96,71,55)(36,54,72,95) );
G=PermutationGroup([[(1,90,17,49),(2,50,18,91),(3,92,19,51),(4,52,20,93),(5,94,21,53),(6,54,22,95),(7,96,23,55),(8,56,24,85),(9,86,13,57),(10,58,14,87),(11,88,15,59),(12,60,16,89),(25,40,61,77),(26,78,62,41),(27,42,63,79),(28,80,64,43),(29,44,65,81),(30,82,66,45),(31,46,67,83),(32,84,68,47),(33,48,69,73),(34,74,70,37),(35,38,71,75),(36,76,72,39)], [(1,39,17,76),(2,25,18,61),(3,41,19,78),(4,27,20,63),(5,43,21,80),(6,29,22,65),(7,45,23,82),(8,31,24,67),(9,47,13,84),(10,33,14,69),(11,37,15,74),(12,35,16,71),(26,92,62,51),(28,94,64,53),(30,96,66,55),(32,86,68,57),(34,88,70,59),(36,90,72,49),(38,60,75,89),(40,50,77,91),(42,52,79,93),(44,54,81,95),(46,56,83,85),(48,58,73,87)], [(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)], [(1,81,17,44),(2,43,18,80),(3,79,19,42),(4,41,20,78),(5,77,21,40),(6,39,22,76),(7,75,23,38),(8,37,24,74),(9,73,13,48),(10,47,14,84),(11,83,15,46),(12,45,16,82),(25,94,61,53),(26,52,62,93),(27,92,63,51),(28,50,64,91),(29,90,65,49),(30,60,66,89),(31,88,67,59),(32,58,68,87),(33,86,69,57),(34,56,70,85),(35,96,71,55),(36,54,72,95)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 12 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 8 | 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 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D4 | D6 | D6 | D6 | D12 | C4○D8 | C8.C22 | S3×D4 | S3×D4 | Q8.7D6 | Q16⋊S3 |
| kernel | Q8.11D12 | C6.D8 | D6⋊C8 | C3×Q8⋊C4 | C4.D12 | C2×C24⋊C2 | C2×C3⋊Q16 | C2×Q8⋊3S3 | Q8⋊C4 | D12 | C2×Dic3 | C3×Q8 | C22×S3 | C4⋊C4 | C2×C8 | C2×Q8 | Q8 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of Q8.11D12 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 71 |
| 0 | 0 | 1 | 72 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 46 | 54 |
| 0 | 0 | 0 | 27 |
| 66 | 7 | 0 | 0 |
| 66 | 59 | 0 | 0 |
| 0 | 0 | 41 | 32 |
| 0 | 0 | 57 | 32 |
| 66 | 7 | 0 | 0 |
| 14 | 7 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 6 | 0 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,71,72],[72,0,0,0,0,72,0,0,0,0,46,0,0,0,54,27],[66,66,0,0,7,59,0,0,0,0,41,57,0,0,32,32],[66,14,0,0,7,7,0,0,0,0,0,6,0,0,12,0] >;
Q8.11D12 in GAP, Magma, Sage, TeX
Q_8._{11}D_{12} % in TeX
G:=Group("Q8.11D12"); // GroupNames label
G:=SmallGroup(192,367);
// by ID
G=gap.SmallGroup(192,367);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,758,135,184,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^12=1,b^2=d^2=a^2,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^-1*b,d*b*d^-1=a*b,d*c*d^-1=a^2*c^-1>;
// generators/relations