metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊1Q8, Dic3.5SD16, C12⋊Q8⋊6C2, C4.3(S3×Q8), C4.Q8⋊11S3, C4⋊C4.164D6, C3⋊3(D4⋊2Q8), (C2×C8).141D6, C12.15(C2×Q8), Dic3⋊C8⋊30C2, C2.25(S3×SD16), C6.41(C2×SD16), C6.D8.5C2, Dic3⋊5D4.5C2, C4.76(C4○D12), C2.24(Q8⋊3D6), C6.73(C8⋊C22), C12.Q8⋊18C2, (C2×Dic3).44D4, C2.D24.13C2, C22.221(S3×D4), C6.37(C22⋊Q8), C12.168(C4○D4), (C2×C24).288C22, (C2×C12).286C23, C2.14(D6⋊Q8), (C2×D12).78C22, C4⋊Dic3.114C22, (C4×Dic3).32C22, (C3×C4.Q8)⋊19C2, (C2×C6).291(C2×D4), (C2×C3⋊C8).63C22, (C3×C4⋊C4).79C22, (C2×C4).389(C22×S3), SmallGroup(192,429)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊Q8
G = < a,b,c,d | a12=b2=c4=1, d2=c2, bab=cac-1=a-1, dad-1=a5, cbc-1=a7b, dbd-1=a10b, dcd-1=c-1 >
Subgroups: 352 in 108 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C2×Q8, C3⋊C8, C24, Dic6, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, D4⋊C4, C4⋊C8, C4.Q8, C4.Q8, C4×D4, C4⋊Q8, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, C2×Dic6, S3×C2×C4, C2×D12, D4⋊2Q8, C12.Q8, C6.D8, Dic3⋊C8, C2.D24, C3×C4.Q8, C12⋊Q8, Dic3⋊5D4, D12⋊Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, SD16, C2×D4, C2×Q8, C4○D4, C22×S3, C22⋊Q8, C2×SD16, C8⋊C22, C4○D12, S3×D4, S3×Q8, D4⋊2Q8, D6⋊Q8, S3×SD16, Q8⋊3D6, D12⋊Q8
►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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 17)(14 16)(18 24)(19 23)(20 22)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(37 44)(38 43)(39 42)(40 41)(45 48)(46 47)(49 54)(50 53)(51 52)(55 60)(56 59)(57 58)(61 67)(62 66)(63 65)(68 72)(69 71)(73 77)(74 76)(78 84)(79 83)(80 82)(85 93)(86 92)(87 91)(88 90)(94 96)
(1 83 52 66)(2 82 53 65)(3 81 54 64)(4 80 55 63)(5 79 56 62)(6 78 57 61)(7 77 58 72)(8 76 59 71)(9 75 60 70)(10 74 49 69)(11 73 50 68)(12 84 51 67)(13 29 87 45)(14 28 88 44)(15 27 89 43)(16 26 90 42)(17 25 91 41)(18 36 92 40)(19 35 93 39)(20 34 94 38)(21 33 95 37)(22 32 96 48)(23 31 85 47)(24 30 86 46)
(1 44 52 28)(2 37 53 33)(3 42 54 26)(4 47 55 31)(5 40 56 36)(6 45 57 29)(7 38 58 34)(8 43 59 27)(9 48 60 32)(10 41 49 25)(11 46 50 30)(12 39 51 35)(13 78 87 61)(14 83 88 66)(15 76 89 71)(16 81 90 64)(17 74 91 69)(18 79 92 62)(19 84 93 67)(20 77 94 72)(21 82 95 65)(22 75 96 70)(23 80 85 63)(24 73 86 68)
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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,17)(14,16)(18,24)(19,23)(20,22)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(37,44)(38,43)(39,42)(40,41)(45,48)(46,47)(49,54)(50,53)(51,52)(55,60)(56,59)(57,58)(61,67)(62,66)(63,65)(68,72)(69,71)(73,77)(74,76)(78,84)(79,83)(80,82)(85,93)(86,92)(87,91)(88,90)(94,96), (1,83,52,66)(2,82,53,65)(3,81,54,64)(4,80,55,63)(5,79,56,62)(6,78,57,61)(7,77,58,72)(8,76,59,71)(9,75,60,70)(10,74,49,69)(11,73,50,68)(12,84,51,67)(13,29,87,45)(14,28,88,44)(15,27,89,43)(16,26,90,42)(17,25,91,41)(18,36,92,40)(19,35,93,39)(20,34,94,38)(21,33,95,37)(22,32,96,48)(23,31,85,47)(24,30,86,46), (1,44,52,28)(2,37,53,33)(3,42,54,26)(4,47,55,31)(5,40,56,36)(6,45,57,29)(7,38,58,34)(8,43,59,27)(9,48,60,32)(10,41,49,25)(11,46,50,30)(12,39,51,35)(13,78,87,61)(14,83,88,66)(15,76,89,71)(16,81,90,64)(17,74,91,69)(18,79,92,62)(19,84,93,67)(20,77,94,72)(21,82,95,65)(22,75,96,70)(23,80,85,63)(24,73,86,68)>;
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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,17)(14,16)(18,24)(19,23)(20,22)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(37,44)(38,43)(39,42)(40,41)(45,48)(46,47)(49,54)(50,53)(51,52)(55,60)(56,59)(57,58)(61,67)(62,66)(63,65)(68,72)(69,71)(73,77)(74,76)(78,84)(79,83)(80,82)(85,93)(86,92)(87,91)(88,90)(94,96), (1,83,52,66)(2,82,53,65)(3,81,54,64)(4,80,55,63)(5,79,56,62)(6,78,57,61)(7,77,58,72)(8,76,59,71)(9,75,60,70)(10,74,49,69)(11,73,50,68)(12,84,51,67)(13,29,87,45)(14,28,88,44)(15,27,89,43)(16,26,90,42)(17,25,91,41)(18,36,92,40)(19,35,93,39)(20,34,94,38)(21,33,95,37)(22,32,96,48)(23,31,85,47)(24,30,86,46), (1,44,52,28)(2,37,53,33)(3,42,54,26)(4,47,55,31)(5,40,56,36)(6,45,57,29)(7,38,58,34)(8,43,59,27)(9,48,60,32)(10,41,49,25)(11,46,50,30)(12,39,51,35)(13,78,87,61)(14,83,88,66)(15,76,89,71)(16,81,90,64)(17,74,91,69)(18,79,92,62)(19,84,93,67)(20,77,94,72)(21,82,95,65)(22,75,96,70)(23,80,85,63)(24,73,86,68) );
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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,17),(14,16),(18,24),(19,23),(20,22),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(37,44),(38,43),(39,42),(40,41),(45,48),(46,47),(49,54),(50,53),(51,52),(55,60),(56,59),(57,58),(61,67),(62,66),(63,65),(68,72),(69,71),(73,77),(74,76),(78,84),(79,83),(80,82),(85,93),(86,92),(87,91),(88,90),(94,96)], [(1,83,52,66),(2,82,53,65),(3,81,54,64),(4,80,55,63),(5,79,56,62),(6,78,57,61),(7,77,58,72),(8,76,59,71),(9,75,60,70),(10,74,49,69),(11,73,50,68),(12,84,51,67),(13,29,87,45),(14,28,88,44),(15,27,89,43),(16,26,90,42),(17,25,91,41),(18,36,92,40),(19,35,93,39),(20,34,94,38),(21,33,95,37),(22,32,96,48),(23,31,85,47),(24,30,86,46)], [(1,44,52,28),(2,37,53,33),(3,42,54,26),(4,47,55,31),(5,40,56,36),(6,45,57,29),(7,38,58,34),(8,43,59,27),(9,48,60,32),(10,41,49,25),(11,46,50,30),(12,39,51,35),(13,78,87,61),(14,83,88,66),(15,76,89,71),(16,81,90,64),(17,74,91,69),(18,79,92,62),(19,84,93,67),(20,77,94,72),(21,82,95,65),(22,75,96,70),(23,80,85,63),(24,73,86,68)]])
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 | SD16 | C4○D4 | C4○D12 | C8⋊C22 | S3×Q8 | S3×D4 | S3×SD16 | Q8⋊3D6 |
| kernel | D12⋊Q8 | C12.Q8 | C6.D8 | Dic3⋊C8 | C2.D24 | C3×C4.Q8 | C12⋊Q8 | Dic3⋊5D4 | C4.Q8 | D12 | C2×Dic3 | C4⋊C4 | C2×C8 | Dic3 | C12 | C4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of D12⋊Q8 ►in GL4(𝔽73) generated by
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 1 |
| 67 | 67 | 0 | 0 |
| 67 | 6 | 0 | 0 |
| 0 | 0 | 13 | 30 |
| 0 | 0 | 43 | 60 |
| 0 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 46 |
| 0 | 0 | 46 | 0 |
G:=sub<GL(4,GF(73))| [0,72,0,0,1,0,0,0,0,0,0,72,0,0,1,1],[0,1,0,0,1,0,0,0,0,0,72,0,0,0,1,1],[67,67,0,0,67,6,0,0,0,0,13,43,0,0,30,60],[0,1,0,0,72,0,0,0,0,0,0,46,0,0,46,0] >;
D12⋊Q8 in GAP, Magma, Sage, TeX
D_{12}\rtimes Q_8 % in TeX
G:=Group("D12:Q8"); // GroupNames label
G:=SmallGroup(192,429);
// by ID
G=gap.SmallGroup(192,429);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,590,219,100,1684,851,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=1,d^2=c^2,b*a*b=c*a*c^-1=a^-1,d*a*d^-1=a^5,c*b*c^-1=a^7*b,d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;
// generators/relations