metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.5D12, C24.84D4, Q8.10D12, M4(2).35D6, C8○D4.3S3, (C2×C8).82D6, C4○D4.49D6, (C3×D4).22D4, C4.21(C2×D12), C12.44(C2×D4), (C3×Q8).22D4, C3⋊3(D4.5D4), C8.41(C3⋊D4), C24.C4⋊16C2, (C2×Dic12)⋊13C2, C6.78(C4⋊D4), (C2×C24).68C22, Q8.14D6.1C2, C12.47D4⋊14C2, C2.26(C12⋊7D4), (C2×C12).423C23, C22.10(C4○D12), C4.Dic3.18C22, (C2×Dic6).118C22, (C3×M4(2)).38C22, (C3×C8○D4).1C2, (C2×C6).8(C4○D4), C4.119(C2×C3⋊D4), (C2×C4).125(C22×S3), (C3×C4○D4).38C22, SmallGroup(192,702)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.10D12
G = < a,b,c,d | a4=1, b2=c12=d2=a2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2c11 >
Subgroups: 256 in 100 conjugacy classes, 39 normal (31 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C6, C6 [×2], C8 [×2], C8 [×3], C2×C4, C2×C4 [×3], D4, D4, Q8, Q8 [×4], Dic3 [×2], C12 [×2], C12, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), M4(2) [×3], SD16 [×2], Q16 [×4], C2×Q8 [×2], C4○D4, C3⋊C8 [×2], C24 [×2], C24, Dic6 [×4], C2×Dic3 [×2], C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C4.10D4 [×2], C8.C4, C8○D4, C2×Q16, C8.C22 [×2], Dic12 [×2], C4.Dic3 [×2], D4.S3 [×2], C3⋊Q16 [×2], C2×C24, C2×C24, C3×M4(2), C3×M4(2), C2×Dic6 [×2], C3×C4○D4, D4.5D4, C24.C4, C12.47D4 [×2], C2×Dic12, Q8.14D6 [×2], C3×C8○D4, Q8.10D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, D12 [×2], C3⋊D4 [×2], C22×S3, C4⋊D4, C2×D12, C4○D12, C2×C3⋊D4, D4.5D4, C12⋊7D4, Q8.10D12
►On 96 points
(1 66 13 54)(2 67 14 55)(3 68 15 56)(4 69 16 57)(5 70 17 58)(6 71 18 59)(7 72 19 60)(8 49 20 61)(9 50 21 62)(10 51 22 63)(11 52 23 64)(12 53 24 65)(25 95 37 83)(26 96 38 84)(27 73 39 85)(28 74 40 86)(29 75 41 87)(30 76 42 88)(31 77 43 89)(32 78 44 90)(33 79 45 91)(34 80 46 92)(35 81 47 93)(36 82 48 94)
(1 7 13 19)(2 8 14 20)(3 9 15 21)(4 10 16 22)(5 11 17 23)(6 12 18 24)(25 77 37 89)(26 78 38 90)(27 79 39 91)(28 80 40 92)(29 81 41 93)(30 82 42 94)(31 83 43 95)(32 84 44 96)(33 85 45 73)(34 86 46 74)(35 87 47 75)(36 88 48 76)(49 67 61 55)(50 68 62 56)(51 69 63 57)(52 70 64 58)(53 71 65 59)(54 72 66 60)
(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 92 13 80)(2 91 14 79)(3 90 15 78)(4 89 16 77)(5 88 17 76)(6 87 18 75)(7 86 19 74)(8 85 20 73)(9 84 21 96)(10 83 22 95)(11 82 23 94)(12 81 24 93)(25 51 37 63)(26 50 38 62)(27 49 39 61)(28 72 40 60)(29 71 41 59)(30 70 42 58)(31 69 43 57)(32 68 44 56)(33 67 45 55)(34 66 46 54)(35 65 47 53)(36 64 48 52)
G:=sub<Sym(96)| (1,66,13,54)(2,67,14,55)(3,68,15,56)(4,69,16,57)(5,70,17,58)(6,71,18,59)(7,72,19,60)(8,49,20,61)(9,50,21,62)(10,51,22,63)(11,52,23,64)(12,53,24,65)(25,95,37,83)(26,96,38,84)(27,73,39,85)(28,74,40,86)(29,75,41,87)(30,76,42,88)(31,77,43,89)(32,78,44,90)(33,79,45,91)(34,80,46,92)(35,81,47,93)(36,82,48,94), (1,7,13,19)(2,8,14,20)(3,9,15,21)(4,10,16,22)(5,11,17,23)(6,12,18,24)(25,77,37,89)(26,78,38,90)(27,79,39,91)(28,80,40,92)(29,81,41,93)(30,82,42,94)(31,83,43,95)(32,84,44,96)(33,85,45,73)(34,86,46,74)(35,87,47,75)(36,88,48,76)(49,67,61,55)(50,68,62,56)(51,69,63,57)(52,70,64,58)(53,71,65,59)(54,72,66,60), (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), (1,92,13,80)(2,91,14,79)(3,90,15,78)(4,89,16,77)(5,88,17,76)(6,87,18,75)(7,86,19,74)(8,85,20,73)(9,84,21,96)(10,83,22,95)(11,82,23,94)(12,81,24,93)(25,51,37,63)(26,50,38,62)(27,49,39,61)(28,72,40,60)(29,71,41,59)(30,70,42,58)(31,69,43,57)(32,68,44,56)(33,67,45,55)(34,66,46,54)(35,65,47,53)(36,64,48,52)>;
G:=Group( (1,66,13,54)(2,67,14,55)(3,68,15,56)(4,69,16,57)(5,70,17,58)(6,71,18,59)(7,72,19,60)(8,49,20,61)(9,50,21,62)(10,51,22,63)(11,52,23,64)(12,53,24,65)(25,95,37,83)(26,96,38,84)(27,73,39,85)(28,74,40,86)(29,75,41,87)(30,76,42,88)(31,77,43,89)(32,78,44,90)(33,79,45,91)(34,80,46,92)(35,81,47,93)(36,82,48,94), (1,7,13,19)(2,8,14,20)(3,9,15,21)(4,10,16,22)(5,11,17,23)(6,12,18,24)(25,77,37,89)(26,78,38,90)(27,79,39,91)(28,80,40,92)(29,81,41,93)(30,82,42,94)(31,83,43,95)(32,84,44,96)(33,85,45,73)(34,86,46,74)(35,87,47,75)(36,88,48,76)(49,67,61,55)(50,68,62,56)(51,69,63,57)(52,70,64,58)(53,71,65,59)(54,72,66,60), (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), (1,92,13,80)(2,91,14,79)(3,90,15,78)(4,89,16,77)(5,88,17,76)(6,87,18,75)(7,86,19,74)(8,85,20,73)(9,84,21,96)(10,83,22,95)(11,82,23,94)(12,81,24,93)(25,51,37,63)(26,50,38,62)(27,49,39,61)(28,72,40,60)(29,71,41,59)(30,70,42,58)(31,69,43,57)(32,68,44,56)(33,67,45,55)(34,66,46,54)(35,65,47,53)(36,64,48,52) );
G=PermutationGroup([(1,66,13,54),(2,67,14,55),(3,68,15,56),(4,69,16,57),(5,70,17,58),(6,71,18,59),(7,72,19,60),(8,49,20,61),(9,50,21,62),(10,51,22,63),(11,52,23,64),(12,53,24,65),(25,95,37,83),(26,96,38,84),(27,73,39,85),(28,74,40,86),(29,75,41,87),(30,76,42,88),(31,77,43,89),(32,78,44,90),(33,79,45,91),(34,80,46,92),(35,81,47,93),(36,82,48,94)], [(1,7,13,19),(2,8,14,20),(3,9,15,21),(4,10,16,22),(5,11,17,23),(6,12,18,24),(25,77,37,89),(26,78,38,90),(27,79,39,91),(28,80,40,92),(29,81,41,93),(30,82,42,94),(31,83,43,95),(32,84,44,96),(33,85,45,73),(34,86,46,74),(35,87,47,75),(36,88,48,76),(49,67,61,55),(50,68,62,56),(51,69,63,57),(52,70,64,58),(53,71,65,59),(54,72,66,60)], [(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)], [(1,92,13,80),(2,91,14,79),(3,90,15,78),(4,89,16,77),(5,88,17,76),(6,87,18,75),(7,86,19,74),(8,85,20,73),(9,84,21,96),(10,83,22,95),(11,82,23,94),(12,81,24,93),(25,51,37,63),(26,50,38,62),(27,49,39,61),(28,72,40,60),(29,71,41,59),(30,70,42,58),(31,69,43,57),(32,68,44,56),(33,67,45,55),(34,66,46,54),(35,65,47,53),(36,64,48,52)])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 12A | 12B | 12C | 12D | 12E | 24A | 24B | 24C | 24D | 24E | ··· | 24J |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 24 | 24 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 24 | 24 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | D12 | D12 | C4○D12 | D4.5D4 | Q8.10D12 |
| kernel | Q8.10D12 | C24.C4 | C12.47D4 | C2×Dic12 | Q8.14D6 | C3×C8○D4 | C8○D4 | C24 | C3×D4 | C3×Q8 | C2×C8 | M4(2) | C4○D4 | C2×C6 | C8 | D4 | Q8 | C22 | C3 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 2 | 4 |
Matrix representation of Q8.10D12 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 1 | 72 | 71 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 1 | 2 |
| 0 | 0 | 0 | 1 | 72 | 72 |
| 64 | 0 | 0 | 0 | 0 | 0 |
| 31 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 57 | 16 | 0 | 0 |
| 0 | 0 | 57 | 57 | 0 | 0 |
| 0 | 0 | 16 | 16 | 41 | 41 |
| 0 | 0 | 57 | 0 | 16 | 0 |
| 14 | 43 | 0 | 0 | 0 | 0 |
| 43 | 59 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 66 | 6 |
| 0 | 0 | 13 | 0 | 60 | 53 |
| 0 | 0 | 60 | 67 | 66 | 6 |
| 0 | 0 | 13 | 66 | 0 | 0 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,72,1,0,0,0,1,0,1,0,0,1,72,0,0,0,0,0,71,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,72,0,0,0,1,0,72,1,0,0,0,0,1,72,0,0,0,0,2,72],[64,31,0,0,0,0,0,8,0,0,0,0,0,0,57,57,16,57,0,0,16,57,16,0,0,0,0,0,41,16,0,0,0,0,41,0],[14,43,0,0,0,0,43,59,0,0,0,0,0,0,7,13,60,13,0,0,0,0,67,66,0,0,66,60,66,0,0,0,6,53,6,0] >;
Q8.10D12 in GAP, Magma, Sage, TeX
Q_8._{10}D_{12} % in TeX
G:=Group("Q8.10D12"); // GroupNames label
G:=SmallGroup(192,702);
// by ID
G=gap.SmallGroup(192,702);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,344,254,1123,297,136,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=1,b^2=c^12=d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=a^2*c^11>;
// generators/relations