metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊10Q8, C42.130D6, C6.1112+ 1+4, (C4×Q8)⋊16S3, D6.8(C2×Q8), C4.50(S3×Q8), C4⋊C4.326D6, D6⋊3Q8⋊8C2, (Q8×C12)⋊14C2, C3⋊4(D4⋊3Q8), D6⋊Q8⋊11C2, C12⋊2Q8⋊28C2, (C4×D12).21C2, (C2×Q8).202D6, C12.108(C2×Q8), C2.23(D4○D12), C4.67(C4○D12), C6.31(C22×Q8), (C2×C6).123C24, C12.6Q8⋊18C2, C12.118(C4○D4), (C4×C12).175C22, (C2×C12).590C23, D6⋊C4.103C22, (C6×Q8).223C22, (C2×D12).289C22, Dic3⋊C4.69C22, C4⋊Dic3.202C22, C22.144(S3×C23), (C2×Dic3).55C23, (C2×Dic6).31C22, (C22×S3).180C23, (S3×C4⋊C4)⋊18C2, C2.14(C2×S3×Q8), C6.55(C2×C4○D4), C2.62(C2×C4○D12), (S3×C2×C4).74C22, (C3×C4⋊C4).351C22, (C2×C4).169(C22×S3), SmallGroup(192,1138)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊10Q8
G = < a,b,c,d | a12=b2=c4=1, d2=c2, bab=a-1, ac=ca, ad=da, cbc-1=a6b, bd=db, dcd-1=c-1 >
Subgroups: 552 in 228 conjugacy classes, 107 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic6, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C2×C4⋊C4, C4×D4, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C6×Q8, D4⋊3Q8, C12⋊2Q8, C12.6Q8, C4×D12, C4×D12, S3×C4⋊C4, D6⋊Q8, D6⋊3Q8, Q8×C12, D12⋊10Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C4○D4, C24, C22×S3, C22×Q8, C2×C4○D4, 2+ 1+4, C4○D12, S3×Q8, S3×C23, D4⋊3Q8, C2×C4○D12, C2×S3×Q8, D4○D12, D12⋊10Q8
►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 3)(4 12)(5 11)(6 10)(7 9)(13 19)(14 18)(15 17)(20 24)(21 23)(25 35)(26 34)(27 33)(28 32)(29 31)(37 39)(40 48)(41 47)(42 46)(43 45)(50 60)(51 59)(52 58)(53 57)(54 56)(61 67)(62 66)(63 65)(68 72)(69 71)(73 81)(74 80)(75 79)(76 78)(82 84)(85 93)(86 92)(87 91)(88 90)(94 96)
(1 57 63 24)(2 58 64 13)(3 59 65 14)(4 60 66 15)(5 49 67 16)(6 50 68 17)(7 51 69 18)(8 52 70 19)(9 53 71 20)(10 54 72 21)(11 55 61 22)(12 56 62 23)(25 87 78 48)(26 88 79 37)(27 89 80 38)(28 90 81 39)(29 91 82 40)(30 92 83 41)(31 93 84 42)(32 94 73 43)(33 95 74 44)(34 96 75 45)(35 85 76 46)(36 86 77 47)
(1 35 63 76)(2 36 64 77)(3 25 65 78)(4 26 66 79)(5 27 67 80)(6 28 68 81)(7 29 69 82)(8 30 70 83)(9 31 71 84)(10 32 72 73)(11 33 61 74)(12 34 62 75)(13 86 58 47)(14 87 59 48)(15 88 60 37)(16 89 49 38)(17 90 50 39)(18 91 51 40)(19 92 52 41)(20 93 53 42)(21 94 54 43)(22 95 55 44)(23 96 56 45)(24 85 57 46)
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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,19)(14,18)(15,17)(20,24)(21,23)(25,35)(26,34)(27,33)(28,32)(29,31)(37,39)(40,48)(41,47)(42,46)(43,45)(50,60)(51,59)(52,58)(53,57)(54,56)(61,67)(62,66)(63,65)(68,72)(69,71)(73,81)(74,80)(75,79)(76,78)(82,84)(85,93)(86,92)(87,91)(88,90)(94,96), (1,57,63,24)(2,58,64,13)(3,59,65,14)(4,60,66,15)(5,49,67,16)(6,50,68,17)(7,51,69,18)(8,52,70,19)(9,53,71,20)(10,54,72,21)(11,55,61,22)(12,56,62,23)(25,87,78,48)(26,88,79,37)(27,89,80,38)(28,90,81,39)(29,91,82,40)(30,92,83,41)(31,93,84,42)(32,94,73,43)(33,95,74,44)(34,96,75,45)(35,85,76,46)(36,86,77,47), (1,35,63,76)(2,36,64,77)(3,25,65,78)(4,26,66,79)(5,27,67,80)(6,28,68,81)(7,29,69,82)(8,30,70,83)(9,31,71,84)(10,32,72,73)(11,33,61,74)(12,34,62,75)(13,86,58,47)(14,87,59,48)(15,88,60,37)(16,89,49,38)(17,90,50,39)(18,91,51,40)(19,92,52,41)(20,93,53,42)(21,94,54,43)(22,95,55,44)(23,96,56,45)(24,85,57,46)>;
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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,19)(14,18)(15,17)(20,24)(21,23)(25,35)(26,34)(27,33)(28,32)(29,31)(37,39)(40,48)(41,47)(42,46)(43,45)(50,60)(51,59)(52,58)(53,57)(54,56)(61,67)(62,66)(63,65)(68,72)(69,71)(73,81)(74,80)(75,79)(76,78)(82,84)(85,93)(86,92)(87,91)(88,90)(94,96), (1,57,63,24)(2,58,64,13)(3,59,65,14)(4,60,66,15)(5,49,67,16)(6,50,68,17)(7,51,69,18)(8,52,70,19)(9,53,71,20)(10,54,72,21)(11,55,61,22)(12,56,62,23)(25,87,78,48)(26,88,79,37)(27,89,80,38)(28,90,81,39)(29,91,82,40)(30,92,83,41)(31,93,84,42)(32,94,73,43)(33,95,74,44)(34,96,75,45)(35,85,76,46)(36,86,77,47), (1,35,63,76)(2,36,64,77)(3,25,65,78)(4,26,66,79)(5,27,67,80)(6,28,68,81)(7,29,69,82)(8,30,70,83)(9,31,71,84)(10,32,72,73)(11,33,61,74)(12,34,62,75)(13,86,58,47)(14,87,59,48)(15,88,60,37)(16,89,49,38)(17,90,50,39)(18,91,51,40)(19,92,52,41)(20,93,53,42)(21,94,54,43)(22,95,55,44)(23,96,56,45)(24,85,57,46) );
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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,19),(14,18),(15,17),(20,24),(21,23),(25,35),(26,34),(27,33),(28,32),(29,31),(37,39),(40,48),(41,47),(42,46),(43,45),(50,60),(51,59),(52,58),(53,57),(54,56),(61,67),(62,66),(63,65),(68,72),(69,71),(73,81),(74,80),(75,79),(76,78),(82,84),(85,93),(86,92),(87,91),(88,90),(94,96)], [(1,57,63,24),(2,58,64,13),(3,59,65,14),(4,60,66,15),(5,49,67,16),(6,50,68,17),(7,51,69,18),(8,52,70,19),(9,53,71,20),(10,54,72,21),(11,55,61,22),(12,56,62,23),(25,87,78,48),(26,88,79,37),(27,89,80,38),(28,90,81,39),(29,91,82,40),(30,92,83,41),(31,93,84,42),(32,94,73,43),(33,95,74,44),(34,96,75,45),(35,85,76,46),(36,86,77,47)], [(1,35,63,76),(2,36,64,77),(3,25,65,78),(4,26,66,79),(5,27,67,80),(6,28,68,81),(7,29,69,82),(8,30,70,83),(9,31,71,84),(10,32,72,73),(11,33,61,74),(12,34,62,75),(13,86,58,47),(14,87,59,48),(15,88,60,37),(16,89,49,38),(17,90,50,39),(18,91,51,40),(19,92,52,41),(20,93,53,42),(21,94,54,43),(22,95,55,44),(23,96,56,45),(24,85,57,46)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | D6 | C4○D4 | C4○D12 | 2+ 1+4 | S3×Q8 | D4○D12 |
| kernel | D12⋊10Q8 | C12⋊2Q8 | C12.6Q8 | C4×D12 | S3×C4⋊C4 | D6⋊Q8 | D6⋊3Q8 | Q8×C12 | C4×Q8 | D12 | C42 | C4⋊C4 | C2×Q8 | C12 | C4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 3 | 2 | 4 | 2 | 1 | 1 | 4 | 3 | 3 | 1 | 4 | 8 | 1 | 2 | 2 |
Matrix representation of D12⋊10Q8 ►in GL4(𝔽13) generated by
| 7 | 3 | 0 | 0 |
| 10 | 10 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 11 | 4 | 0 | 0 |
| 9 | 2 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 9 |
| 0 | 0 | 9 | 10 |
G:=sub<GL(4,GF(13))| [7,10,0,0,3,10,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,1,1,0,0,0,0,1,0,0,0,0,1],[11,9,0,0,4,2,0,0,0,0,0,1,0,0,12,0],[1,0,0,0,0,1,0,0,0,0,3,9,0,0,9,10] >;
D12⋊10Q8 in GAP, Magma, Sage, TeX
D_{12}\rtimes_{10}Q_8 % in TeX
G:=Group("D12:10Q8"); // GroupNames label
G:=SmallGroup(192,1138);
// by ID
G=gap.SmallGroup(192,1138);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,232,100,185,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^6*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations