metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊6D12, C42.112D6, C6.612- 1+4, (C3×D4)⋊11D4, (C4×D4)⋊17S3, (D4×C12)⋊19C2, (C4×D12)⋊31C2, C3⋊3(D4⋊6D4), C4⋊C4.284D6, C12.55(C2×D4), C4.23(C2×D12), C12⋊14(C4○D4), C4⋊5(D4⋊2S3), C12⋊7D4⋊10C2, C4.D12⋊15C2, C12⋊2Q8⋊25C2, (C2×D4).249D6, (C2×C6).99C24, D6⋊C4.5C22, C6.17(C22×D4), C22.2(C2×D12), C22⋊C4.113D6, C2.18(Q8○D12), C2.19(C22×D12), (C22×C4).227D6, (C4×C12).155C22, (C2×C12).160C23, (C6×D4).260C22, C23.21D6⋊6C2, C4⋊Dic3.39C22, (C2×D12).212C22, (C22×S3).34C23, C23.183(C22×S3), (C22×C6).169C23, (C22×C12).81C22, C22.124(S3×C23), (C2×Dic6).144C22, (C2×Dic3).206C23, (C22×Dic3).97C22, (C2×C6).2(C2×D4), C6.74(C2×C4○D4), (C2×D4⋊2S3)⋊4C2, (C2×C4⋊Dic3)⋊25C2, (S3×C2×C4).65C22, C2.22(C2×D4⋊2S3), (C3×C4⋊C4).329C22, (C2×C4).732(C22×S3), (C2×C3⋊D4).15C22, (C3×C22⋊C4).106C22, SmallGroup(192,1114)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊6D12
G = < a,b,c,d | a4=b2=c12=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 728 in 292 conjugacy classes, 115 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C2×C4⋊C4, C4×D4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C4⋊Q8, C2×C4○D4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, D4⋊2S3, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, D4⋊6D4, C12⋊2Q8, C4×D12, C23.21D6, C4.D12, C2×C4⋊Dic3, C12⋊7D4, D4×C12, C2×D4⋊2S3, D4⋊6D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, D12, C22×S3, C22×D4, C2×C4○D4, 2- 1+4, C2×D12, D4⋊2S3, S3×C23, D4⋊6D4, C22×D12, C2×D4⋊2S3, Q8○D12, D4⋊6D12
►On 96 points
(1 69 32 48)(2 70 33 37)(3 71 34 38)(4 72 35 39)(5 61 36 40)(6 62 25 41)(7 63 26 42)(8 64 27 43)(9 65 28 44)(10 66 29 45)(11 67 30 46)(12 68 31 47)(13 81 60 96)(14 82 49 85)(15 83 50 86)(16 84 51 87)(17 73 52 88)(18 74 53 89)(19 75 54 90)(20 76 55 91)(21 77 56 92)(22 78 57 93)(23 79 58 94)(24 80 59 95)
(1 91)(2 92)(3 93)(4 94)(5 95)(6 96)(7 85)(8 86)(9 87)(10 88)(11 89)(12 90)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 37)(22 38)(23 39)(24 40)(25 81)(26 82)(27 83)(28 84)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(49 63)(50 64)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)(57 71)(58 72)(59 61)(60 62)
(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 49)(14 60)(15 59)(16 58)(17 57)(18 56)(19 55)(20 54)(21 53)(22 52)(23 51)(24 50)(25 26)(27 36)(28 35)(29 34)(30 33)(31 32)(37 46)(38 45)(39 44)(40 43)(41 42)(47 48)(61 64)(62 63)(65 72)(66 71)(67 70)(68 69)(73 93)(74 92)(75 91)(76 90)(77 89)(78 88)(79 87)(80 86)(81 85)(82 96)(83 95)(84 94)
G:=sub<Sym(96)| (1,69,32,48)(2,70,33,37)(3,71,34,38)(4,72,35,39)(5,61,36,40)(6,62,25,41)(7,63,26,42)(8,64,27,43)(9,65,28,44)(10,66,29,45)(11,67,30,46)(12,68,31,47)(13,81,60,96)(14,82,49,85)(15,83,50,86)(16,84,51,87)(17,73,52,88)(18,74,53,89)(19,75,54,90)(20,76,55,91)(21,77,56,92)(22,78,57,93)(23,79,58,94)(24,80,59,95), (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,37)(22,38)(23,39)(24,40)(25,81)(26,82)(27,83)(28,84)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(49,63)(50,64)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70)(57,71)(58,72)(59,61)(60,62), (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)(13,49)(14,60)(15,59)(16,58)(17,57)(18,56)(19,55)(20,54)(21,53)(22,52)(23,51)(24,50)(25,26)(27,36)(28,35)(29,34)(30,33)(31,32)(37,46)(38,45)(39,44)(40,43)(41,42)(47,48)(61,64)(62,63)(65,72)(66,71)(67,70)(68,69)(73,93)(74,92)(75,91)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,96)(83,95)(84,94)>;
G:=Group( (1,69,32,48)(2,70,33,37)(3,71,34,38)(4,72,35,39)(5,61,36,40)(6,62,25,41)(7,63,26,42)(8,64,27,43)(9,65,28,44)(10,66,29,45)(11,67,30,46)(12,68,31,47)(13,81,60,96)(14,82,49,85)(15,83,50,86)(16,84,51,87)(17,73,52,88)(18,74,53,89)(19,75,54,90)(20,76,55,91)(21,77,56,92)(22,78,57,93)(23,79,58,94)(24,80,59,95), (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,37)(22,38)(23,39)(24,40)(25,81)(26,82)(27,83)(28,84)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(49,63)(50,64)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70)(57,71)(58,72)(59,61)(60,62), (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)(13,49)(14,60)(15,59)(16,58)(17,57)(18,56)(19,55)(20,54)(21,53)(22,52)(23,51)(24,50)(25,26)(27,36)(28,35)(29,34)(30,33)(31,32)(37,46)(38,45)(39,44)(40,43)(41,42)(47,48)(61,64)(62,63)(65,72)(66,71)(67,70)(68,69)(73,93)(74,92)(75,91)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,96)(83,95)(84,94) );
G=PermutationGroup([[(1,69,32,48),(2,70,33,37),(3,71,34,38),(4,72,35,39),(5,61,36,40),(6,62,25,41),(7,63,26,42),(8,64,27,43),(9,65,28,44),(10,66,29,45),(11,67,30,46),(12,68,31,47),(13,81,60,96),(14,82,49,85),(15,83,50,86),(16,84,51,87),(17,73,52,88),(18,74,53,89),(19,75,54,90),(20,76,55,91),(21,77,56,92),(22,78,57,93),(23,79,58,94),(24,80,59,95)], [(1,91),(2,92),(3,93),(4,94),(5,95),(6,96),(7,85),(8,86),(9,87),(10,88),(11,89),(12,90),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,37),(22,38),(23,39),(24,40),(25,81),(26,82),(27,83),(28,84),(29,73),(30,74),(31,75),(32,76),(33,77),(34,78),(35,79),(36,80),(49,63),(50,64),(51,65),(52,66),(53,67),(54,68),(55,69),(56,70),(57,71),(58,72),(59,61),(60,62)], [(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),(13,49),(14,60),(15,59),(16,58),(17,57),(18,56),(19,55),(20,54),(21,53),(22,52),(23,51),(24,50),(25,26),(27,36),(28,35),(29,34),(30,33),(31,32),(37,46),(38,45),(39,44),(40,43),(41,42),(47,48),(61,64),(62,63),(65,72),(66,71),(67,70),(68,69),(73,93),(74,92),(75,91),(76,90),(77,89),(78,88),(79,87),(80,86),(81,85),(82,96),(83,95),(84,94)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | D6 | C4○D4 | D12 | 2- 1+4 | D4⋊2S3 | Q8○D12 |
| kernel | D4⋊6D12 | C12⋊2Q8 | C4×D12 | C23.21D6 | C4.D12 | C2×C4⋊Dic3 | C12⋊7D4 | D4×C12 | C2×D4⋊2S3 | C4×D4 | C3×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C12 | D4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 2 | 1 | 4 | 1 | 2 | 1 | 2 | 1 | 4 | 8 | 1 | 2 | 2 |
Matrix representation of D4⋊6D12 ►in GL4(𝔽13) generated by
| 5 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 8 | 0 | 0 |
| 5 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 10 |
| 0 | 0 | 3 | 6 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 3 | 10 |
| 0 | 0 | 7 | 10 |
G:=sub<GL(4,GF(13))| [5,0,0,0,0,8,0,0,0,0,12,0,0,0,0,12],[0,5,0,0,8,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,3,3,0,0,10,6],[1,0,0,0,0,12,0,0,0,0,3,7,0,0,10,10] >;
D4⋊6D12 in GAP, Magma, Sage, TeX
D_4\rtimes_6D_{12} % in TeX
G:=Group("D4:6D12"); // GroupNames label
G:=SmallGroup(192,1114);
// by ID
G=gap.SmallGroup(192,1114);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,387,675,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^12=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations