metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊2Q8, Dic3.7D8, C12⋊Q8⋊7C2, C2.D8⋊8S3, C4.7(S3×Q8), (C2×C8).31D6, C6.31(C2×D8), C2.15(S3×D8), C4⋊C4.173D6, C3⋊3(D4⋊Q8), C12.23(C2×Q8), Dic3⋊C8⋊15C2, C6.Q16⋊23C2, C2.D24.6C2, C6.D8.8C2, Dic3⋊5D4.8C2, C4.84(C4○D12), (C2×Dic3).51D4, C22.235(S3×D4), C6.41(C22⋊Q8), C12.172(C4○D4), (C2×C12).306C23, (C2×C24).173C22, C2.18(D6⋊Q8), C2.25(Q16⋊S3), (C2×D12).86C22, C6.74(C8.C22), C4⋊Dic3.128C22, (C4×Dic3).38C22, (C3×C2.D8)⋊15C2, (C2×C6).311(C2×D4), (C2×C3⋊C8).75C22, (C3×C4⋊C4).99C22, (C2×C4).409(C22×S3), SmallGroup(192,449)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊2Q8
G = < a,b,c,d | a12=b2=c4=1, d2=c2, bab=cac-1=a-1, dad-1=a5, cbc-1=ab, dbd-1=a4b, 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, C2.D8, C2.D8, 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⋊Q8, C6.Q16, C6.D8, Dic3⋊C8, C2.D24, C3×C2.D8, C12⋊Q8, Dic3⋊5D4, D12⋊2Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, D8, C2×D4, C2×Q8, C4○D4, C22×S3, C22⋊Q8, C2×D8, C8.C22, C4○D12, S3×D4, S3×Q8, D4⋊Q8, D6⋊Q8, S3×D8, Q16⋊S3, D12⋊2Q8
►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 15)(16 24)(17 23)(18 22)(19 21)(25 31)(26 30)(27 29)(32 36)(33 35)(37 44)(38 43)(39 42)(40 41)(45 48)(46 47)(50 60)(51 59)(52 58)(53 57)(54 56)(61 64)(62 63)(65 72)(66 71)(67 70)(68 69)(74 84)(75 83)(76 82)(77 81)(78 80)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)
(1 19 69 84)(2 18 70 83)(3 17 71 82)(4 16 72 81)(5 15 61 80)(6 14 62 79)(7 13 63 78)(8 24 64 77)(9 23 65 76)(10 22 66 75)(11 21 67 74)(12 20 68 73)(25 37 52 87)(26 48 53 86)(27 47 54 85)(28 46 55 96)(29 45 56 95)(30 44 57 94)(31 43 58 93)(32 42 59 92)(33 41 60 91)(34 40 49 90)(35 39 50 89)(36 38 51 88)
(1 47 69 85)(2 40 70 90)(3 45 71 95)(4 38 72 88)(5 43 61 93)(6 48 62 86)(7 41 63 91)(8 46 64 96)(9 39 65 89)(10 44 66 94)(11 37 67 87)(12 42 68 92)(13 33 78 60)(14 26 79 53)(15 31 80 58)(16 36 81 51)(17 29 82 56)(18 34 83 49)(19 27 84 54)(20 32 73 59)(21 25 74 52)(22 30 75 57)(23 35 76 50)(24 28 77 55)
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)(13,15)(16,24)(17,23)(18,22)(19,21)(25,31)(26,30)(27,29)(32,36)(33,35)(37,44)(38,43)(39,42)(40,41)(45,48)(46,47)(50,60)(51,59)(52,58)(53,57)(54,56)(61,64)(62,63)(65,72)(66,71)(67,70)(68,69)(74,84)(75,83)(76,82)(77,81)(78,80)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91), (1,19,69,84)(2,18,70,83)(3,17,71,82)(4,16,72,81)(5,15,61,80)(6,14,62,79)(7,13,63,78)(8,24,64,77)(9,23,65,76)(10,22,66,75)(11,21,67,74)(12,20,68,73)(25,37,52,87)(26,48,53,86)(27,47,54,85)(28,46,55,96)(29,45,56,95)(30,44,57,94)(31,43,58,93)(32,42,59,92)(33,41,60,91)(34,40,49,90)(35,39,50,89)(36,38,51,88), (1,47,69,85)(2,40,70,90)(3,45,71,95)(4,38,72,88)(5,43,61,93)(6,48,62,86)(7,41,63,91)(8,46,64,96)(9,39,65,89)(10,44,66,94)(11,37,67,87)(12,42,68,92)(13,33,78,60)(14,26,79,53)(15,31,80,58)(16,36,81,51)(17,29,82,56)(18,34,83,49)(19,27,84,54)(20,32,73,59)(21,25,74,52)(22,30,75,57)(23,35,76,50)(24,28,77,55)>;
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)(13,15)(16,24)(17,23)(18,22)(19,21)(25,31)(26,30)(27,29)(32,36)(33,35)(37,44)(38,43)(39,42)(40,41)(45,48)(46,47)(50,60)(51,59)(52,58)(53,57)(54,56)(61,64)(62,63)(65,72)(66,71)(67,70)(68,69)(74,84)(75,83)(76,82)(77,81)(78,80)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91), (1,19,69,84)(2,18,70,83)(3,17,71,82)(4,16,72,81)(5,15,61,80)(6,14,62,79)(7,13,63,78)(8,24,64,77)(9,23,65,76)(10,22,66,75)(11,21,67,74)(12,20,68,73)(25,37,52,87)(26,48,53,86)(27,47,54,85)(28,46,55,96)(29,45,56,95)(30,44,57,94)(31,43,58,93)(32,42,59,92)(33,41,60,91)(34,40,49,90)(35,39,50,89)(36,38,51,88), (1,47,69,85)(2,40,70,90)(3,45,71,95)(4,38,72,88)(5,43,61,93)(6,48,62,86)(7,41,63,91)(8,46,64,96)(9,39,65,89)(10,44,66,94)(11,37,67,87)(12,42,68,92)(13,33,78,60)(14,26,79,53)(15,31,80,58)(16,36,81,51)(17,29,82,56)(18,34,83,49)(19,27,84,54)(20,32,73,59)(21,25,74,52)(22,30,75,57)(23,35,76,50)(24,28,77,55) );
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),(13,15),(16,24),(17,23),(18,22),(19,21),(25,31),(26,30),(27,29),(32,36),(33,35),(37,44),(38,43),(39,42),(40,41),(45,48),(46,47),(50,60),(51,59),(52,58),(53,57),(54,56),(61,64),(62,63),(65,72),(66,71),(67,70),(68,69),(74,84),(75,83),(76,82),(77,81),(78,80),(85,96),(86,95),(87,94),(88,93),(89,92),(90,91)], [(1,19,69,84),(2,18,70,83),(3,17,71,82),(4,16,72,81),(5,15,61,80),(6,14,62,79),(7,13,63,78),(8,24,64,77),(9,23,65,76),(10,22,66,75),(11,21,67,74),(12,20,68,73),(25,37,52,87),(26,48,53,86),(27,47,54,85),(28,46,55,96),(29,45,56,95),(30,44,57,94),(31,43,58,93),(32,42,59,92),(33,41,60,91),(34,40,49,90),(35,39,50,89),(36,38,51,88)], [(1,47,69,85),(2,40,70,90),(3,45,71,95),(4,38,72,88),(5,43,61,93),(6,48,62,86),(7,41,63,91),(8,46,64,96),(9,39,65,89),(10,44,66,94),(11,37,67,87),(12,42,68,92),(13,33,78,60),(14,26,79,53),(15,31,80,58),(16,36,81,51),(17,29,82,56),(18,34,83,49),(19,27,84,54),(20,32,73,59),(21,25,74,52),(22,30,75,57),(23,35,76,50),(24,28,77,55)]])
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 | D8 | C4○D4 | C4○D12 | C8.C22 | S3×Q8 | S3×D4 | S3×D8 | Q16⋊S3 |
| kernel | D12⋊2Q8 | C6.Q16 | C6.D8 | Dic3⋊C8 | C2.D24 | C3×C2.D8 | C12⋊Q8 | Dic3⋊5D4 | C2.D8 | 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⋊2Q8 ►in GL6(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 16 | 57 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 0 | 46 |
G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,0,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1],[16,16,0,0,0,0,16,57,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,27,0,0,0,0,0,0,46] >;
D12⋊2Q8 in GAP, Magma, Sage, TeX
D_{12}\rtimes_2Q_8 % in TeX
G:=Group("D12:2Q8"); // GroupNames label
G:=SmallGroup(192,449);
// by ID
G=gap.SmallGroup(192,449);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,590,219,268,1684,851,102,6278]);
// 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*b,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations