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, C3, C4, C4, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Dic3, C12, C12, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), M4(2), SD16, Q16, C2×Q8, C4○D4, C3⋊C8, C24, C24, Dic6, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C4.10D4, C8.C4, C8○D4, C2×Q16, C8.C22, Dic12, C4.Dic3, D4.S3, C3⋊Q16, C2×C24, C2×C24, C3×M4(2), C3×M4(2), C2×Dic6, C3×C4○D4, D4.5D4, C24.C4, C12.47D4, C2×Dic12, Q8.14D6, C3×C8○D4, Q8.10D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C3⋊D4, C22×S3, C4⋊D4, C2×D12, C4○D12, C2×C3⋊D4, D4.5D4, C12⋊7D4, Q8.10D12
►On 96 points
Generators in S96
(1 58 13 70)(2 59 14 71)(3 60 15 72)(4 61 16 49)(5 62 17 50)(6 63 18 51)(7 64 19 52)(8 65 20 53)(9 66 21 54)(10 67 22 55)(11 68 23 56)(12 69 24 57)(25 91 37 79)(26 92 38 80)(27 93 39 81)(28 94 40 82)(29 95 41 83)(30 96 42 84)(31 73 43 85)(32 74 44 86)(33 75 45 87)(34 76 46 88)(35 77 47 89)(36 78 48 90)
(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 73 37 85)(26 74 38 86)(27 75 39 87)(28 76 40 88)(29 77 41 89)(30 78 42 90)(31 79 43 91)(32 80 44 92)(33 81 45 93)(34 82 46 94)(35 83 47 95)(36 84 48 96)(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 85 13 73)(2 84 14 96)(3 83 15 95)(4 82 16 94)(5 81 17 93)(6 80 18 92)(7 79 19 91)(8 78 20 90)(9 77 21 89)(10 76 22 88)(11 75 23 87)(12 74 24 86)(25 64 37 52)(26 63 38 51)(27 62 39 50)(28 61 40 49)(29 60 41 72)(30 59 42 71)(31 58 43 70)(32 57 44 69)(33 56 45 68)(34 55 46 67)(35 54 47 66)(36 53 48 65)
G:=sub<Sym(96)| (1,58,13,70)(2,59,14,71)(3,60,15,72)(4,61,16,49)(5,62,17,50)(6,63,18,51)(7,64,19,52)(8,65,20,53)(9,66,21,54)(10,67,22,55)(11,68,23,56)(12,69,24,57)(25,91,37,79)(26,92,38,80)(27,93,39,81)(28,94,40,82)(29,95,41,83)(30,96,42,84)(31,73,43,85)(32,74,44,86)(33,75,45,87)(34,76,46,88)(35,77,47,89)(36,78,48,90), (1,7,13,19)(2,8,14,20)(3,9,15,21)(4,10,16,22)(5,11,17,23)(6,12,18,24)(25,73,37,85)(26,74,38,86)(27,75,39,87)(28,76,40,88)(29,77,41,89)(30,78,42,90)(31,79,43,91)(32,80,44,92)(33,81,45,93)(34,82,46,94)(35,83,47,95)(36,84,48,96)(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), (1,85,13,73)(2,84,14,96)(3,83,15,95)(4,82,16,94)(5,81,17,93)(6,80,18,92)(7,79,19,91)(8,78,20,90)(9,77,21,89)(10,76,22,88)(11,75,23,87)(12,74,24,86)(25,64,37,52)(26,63,38,51)(27,62,39,50)(28,61,40,49)(29,60,41,72)(30,59,42,71)(31,58,43,70)(32,57,44,69)(33,56,45,68)(34,55,46,67)(35,54,47,66)(36,53,48,65)>;
G:=Group( (1,58,13,70)(2,59,14,71)(3,60,15,72)(4,61,16,49)(5,62,17,50)(6,63,18,51)(7,64,19,52)(8,65,20,53)(9,66,21,54)(10,67,22,55)(11,68,23,56)(12,69,24,57)(25,91,37,79)(26,92,38,80)(27,93,39,81)(28,94,40,82)(29,95,41,83)(30,96,42,84)(31,73,43,85)(32,74,44,86)(33,75,45,87)(34,76,46,88)(35,77,47,89)(36,78,48,90), (1,7,13,19)(2,8,14,20)(3,9,15,21)(4,10,16,22)(5,11,17,23)(6,12,18,24)(25,73,37,85)(26,74,38,86)(27,75,39,87)(28,76,40,88)(29,77,41,89)(30,78,42,90)(31,79,43,91)(32,80,44,92)(33,81,45,93)(34,82,46,94)(35,83,47,95)(36,84,48,96)(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), (1,85,13,73)(2,84,14,96)(3,83,15,95)(4,82,16,94)(5,81,17,93)(6,80,18,92)(7,79,19,91)(8,78,20,90)(9,77,21,89)(10,76,22,88)(11,75,23,87)(12,74,24,86)(25,64,37,52)(26,63,38,51)(27,62,39,50)(28,61,40,49)(29,60,41,72)(30,59,42,71)(31,58,43,70)(32,57,44,69)(33,56,45,68)(34,55,46,67)(35,54,47,66)(36,53,48,65) );
G=PermutationGroup([[(1,58,13,70),(2,59,14,71),(3,60,15,72),(4,61,16,49),(5,62,17,50),(6,63,18,51),(7,64,19,52),(8,65,20,53),(9,66,21,54),(10,67,22,55),(11,68,23,56),(12,69,24,57),(25,91,37,79),(26,92,38,80),(27,93,39,81),(28,94,40,82),(29,95,41,83),(30,96,42,84),(31,73,43,85),(32,74,44,86),(33,75,45,87),(34,76,46,88),(35,77,47,89),(36,78,48,90)], [(1,7,13,19),(2,8,14,20),(3,9,15,21),(4,10,16,22),(5,11,17,23),(6,12,18,24),(25,73,37,85),(26,74,38,86),(27,75,39,87),(28,76,40,88),(29,77,41,89),(30,78,42,90),(31,79,43,91),(32,80,44,92),(33,81,45,93),(34,82,46,94),(35,83,47,95),(36,84,48,96),(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)], [(1,85,13,73),(2,84,14,96),(3,83,15,95),(4,82,16,94),(5,81,17,93),(6,80,18,92),(7,79,19,91),(8,78,20,90),(9,77,21,89),(10,76,22,88),(11,75,23,87),(12,74,24,86),(25,64,37,52),(26,63,38,51),(27,62,39,50),(28,61,40,49),(29,60,41,72),(30,59,42,71),(31,58,43,70),(32,57,44,69),(33,56,45,68),(34,55,46,67),(35,54,47,66),(36,53,48,65)]])
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