metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊24D4, C42.109D6, C6.592- 1+4, (C4×D4)⋊13S3, D6⋊3D4⋊8C2, (C4×D12)⋊29C2, (D4×C12)⋊15C2, C12⋊2(C4○D4), C4⋊3(C4○D12), C3⋊2(D4⋊6D4), C4⋊C4.316D6, D6.15(C2×D4), C4.140(S3×D4), C12⋊2Q8⋊24C2, (C2×D4).214D6, C23.9D6⋊6C2, C12.346(C2×D4), (C2×C6).95C24, C6.50(C22×D4), D6⋊C4.98C22, C22⋊C4.110D6, C2.16(Q8○D12), (C22×C4).224D6, C12.48D4⋊20C2, (C4×C12).152C22, (C2×C12).783C23, (C6×D4).257C22, (C2×D12).287C22, Dic3⋊C4.65C22, C4⋊Dic3.199C22, C22.120(S3×C23), C23.105(C22×S3), (C22×C6).165C23, (C2×Dic3).41C23, (C22×S3).173C23, (C22×C12).107C22, (C2×Dic6).239C22, C6.D4.12C22, (S3×C4⋊C4)⋊15C2, C2.23(C2×S3×D4), (C2×C4○D12)⋊8C2, C6.42(C2×C4○D4), C2.46(C2×C4○D12), (S3×C2×C4).64C22, (C3×C4⋊C4).326C22, (C2×C4).579(C22×S3), (C2×C3⋊D4).113C22, (C3×C22⋊C4).122C22, SmallGroup(192,1110)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊24D4
G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a6b, dcd=c-1 >
Subgroups: 728 in 292 conjugacy classes, 107 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, D6, 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, 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, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, C6×D4, D4⋊6D4, C12⋊2Q8, C4×D12, C23.9D6, S3×C4⋊C4, C12.48D4, D6⋊3D4, D4×C12, C2×C4○D12, D12⋊24D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C22×S3, C22×D4, C2×C4○D4, 2- 1+4, C4○D12, S3×D4, S3×C23, D4⋊6D4, C2×C4○D12, C2×S3×D4, Q8○D12, D12⋊24D4
►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 17)(14 16)(18 24)(19 23)(20 22)(25 27)(28 36)(29 35)(30 34)(31 33)(37 43)(38 42)(39 41)(44 48)(45 47)(49 55)(50 54)(51 53)(56 60)(57 59)(61 71)(62 70)(63 69)(64 68)(65 67)(73 79)(74 78)(75 77)(80 84)(81 83)(86 96)(87 95)(88 94)(89 93)(90 92)
(1 20 31 81)(2 21 32 82)(3 22 33 83)(4 23 34 84)(5 24 35 73)(6 13 36 74)(7 14 25 75)(8 15 26 76)(9 16 27 77)(10 17 28 78)(11 18 29 79)(12 19 30 80)(37 88 55 63)(38 89 56 64)(39 90 57 65)(40 91 58 66)(41 92 59 67)(42 93 60 68)(43 94 49 69)(44 95 50 70)(45 96 51 71)(46 85 52 72)(47 86 53 61)(48 87 54 62)
(1 87)(2 88)(3 89)(4 90)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(11 85)(12 86)(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 68)(26 69)(27 70)(28 71)(29 72)(30 61)(31 62)(32 63)(33 64)(34 65)(35 66)(36 67)(49 76)(50 77)(51 78)(52 79)(53 80)(54 81)(55 82)(56 83)(57 84)(58 73)(59 74)(60 75)
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)(13,17)(14,16)(18,24)(19,23)(20,22)(25,27)(28,36)(29,35)(30,34)(31,33)(37,43)(38,42)(39,41)(44,48)(45,47)(49,55)(50,54)(51,53)(56,60)(57,59)(61,71)(62,70)(63,69)(64,68)(65,67)(73,79)(74,78)(75,77)(80,84)(81,83)(86,96)(87,95)(88,94)(89,93)(90,92), (1,20,31,81)(2,21,32,82)(3,22,33,83)(4,23,34,84)(5,24,35,73)(6,13,36,74)(7,14,25,75)(8,15,26,76)(9,16,27,77)(10,17,28,78)(11,18,29,79)(12,19,30,80)(37,88,55,63)(38,89,56,64)(39,90,57,65)(40,91,58,66)(41,92,59,67)(42,93,60,68)(43,94,49,69)(44,95,50,70)(45,96,51,71)(46,85,52,72)(47,86,53,61)(48,87,54,62), (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,85)(12,86)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,37)(22,38)(23,39)(24,40)(25,68)(26,69)(27,70)(28,71)(29,72)(30,61)(31,62)(32,63)(33,64)(34,65)(35,66)(36,67)(49,76)(50,77)(51,78)(52,79)(53,80)(54,81)(55,82)(56,83)(57,84)(58,73)(59,74)(60,75)>;
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)(13,17)(14,16)(18,24)(19,23)(20,22)(25,27)(28,36)(29,35)(30,34)(31,33)(37,43)(38,42)(39,41)(44,48)(45,47)(49,55)(50,54)(51,53)(56,60)(57,59)(61,71)(62,70)(63,69)(64,68)(65,67)(73,79)(74,78)(75,77)(80,84)(81,83)(86,96)(87,95)(88,94)(89,93)(90,92), (1,20,31,81)(2,21,32,82)(3,22,33,83)(4,23,34,84)(5,24,35,73)(6,13,36,74)(7,14,25,75)(8,15,26,76)(9,16,27,77)(10,17,28,78)(11,18,29,79)(12,19,30,80)(37,88,55,63)(38,89,56,64)(39,90,57,65)(40,91,58,66)(41,92,59,67)(42,93,60,68)(43,94,49,69)(44,95,50,70)(45,96,51,71)(46,85,52,72)(47,86,53,61)(48,87,54,62), (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,85)(12,86)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,37)(22,38)(23,39)(24,40)(25,68)(26,69)(27,70)(28,71)(29,72)(30,61)(31,62)(32,63)(33,64)(34,65)(35,66)(36,67)(49,76)(50,77)(51,78)(52,79)(53,80)(54,81)(55,82)(56,83)(57,84)(58,73)(59,74)(60,75) );
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),(13,17),(14,16),(18,24),(19,23),(20,22),(25,27),(28,36),(29,35),(30,34),(31,33),(37,43),(38,42),(39,41),(44,48),(45,47),(49,55),(50,54),(51,53),(56,60),(57,59),(61,71),(62,70),(63,69),(64,68),(65,67),(73,79),(74,78),(75,77),(80,84),(81,83),(86,96),(87,95),(88,94),(89,93),(90,92)], [(1,20,31,81),(2,21,32,82),(3,22,33,83),(4,23,34,84),(5,24,35,73),(6,13,36,74),(7,14,25,75),(8,15,26,76),(9,16,27,77),(10,17,28,78),(11,18,29,79),(12,19,30,80),(37,88,55,63),(38,89,56,64),(39,90,57,65),(40,91,58,66),(41,92,59,67),(42,93,60,68),(43,94,49,69),(44,95,50,70),(45,96,51,71),(46,85,52,72),(47,86,53,61),(48,87,54,62)], [(1,87),(2,88),(3,89),(4,90),(5,91),(6,92),(7,93),(8,94),(9,95),(10,96),(11,85),(12,86),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,37),(22,38),(23,39),(24,40),(25,68),(26,69),(27,70),(28,71),(29,72),(30,61),(31,62),(32,63),(33,64),(34,65),(35,66),(36,67),(49,76),(50,77),(51,78),(52,79),(53,80),(54,81),(55,82),(56,83),(57,84),(58,73),(59,74),(60,75)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | ··· | 4H | 4I | 4J | ··· | 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 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | ··· | 2 | 4 | 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 | C4○D12 | 2- 1+4 | S3×D4 | Q8○D12 |
| kernel | D12⋊24D4 | C12⋊2Q8 | C4×D12 | C23.9D6 | S3×C4⋊C4 | C12.48D4 | D6⋊3D4 | D4×C12 | C2×C4○D12 | C4×D4 | D12 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C12 | C4 | 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 D12⋊24D4 ►in GL4(𝔽13) generated by
| 3 | 10 | 0 | 0 |
| 3 | 6 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 12 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 9 | 11 |
| 0 | 0 | 2 | 4 |
| 2 | 4 | 0 | 0 |
| 9 | 11 | 0 | 0 |
| 0 | 0 | 2 | 4 |
| 0 | 0 | 9 | 11 |
G:=sub<GL(4,GF(13))| [3,3,0,0,10,6,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,12,1,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,9,2,0,0,11,4],[2,9,0,0,4,11,0,0,0,0,2,9,0,0,4,11] >;
D12⋊24D4 in GAP, Magma, Sage, TeX
D_{12}\rtimes_{24}D_4 % in TeX
G:=Group("D12:24D4"); // GroupNames label
G:=SmallGroup(192,1110);
// by ID
G=gap.SmallGroup(192,1110);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,100,675,570,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations