metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊3D12, Dic6⋊3D4, (C3×D4)⋊2D4, D6⋊C8⋊12C2, C4⋊C4.12D6, C4.3(C2×D12), C4.86(S3×D4), C12.2(C2×D4), C12⋊D4⋊3C2, C3⋊2(D4⋊D4), (C2×C8).117D6, D4⋊C4⋊13S3, C6.21C22≀C2, (C2×D4).136D6, C6.42(C4○D8), C6.SD16⋊6C2, C2.17(D8⋊S3), C6.35(C8⋊C22), (C22×S3).13D4, (C6×D4).42C22, C22.179(S3×D4), C2.24(D6⋊D4), (C2×C12).221C23, (C2×C24).128C22, (C2×Dic3).142D4, (C2×D12).51C22, C2.12(Q8.7D6), (C2×Dic6).59C22, (C2×D4⋊S3)⋊4C2, (C2×C24⋊C2)⋊15C2, (C2×D4⋊2S3)⋊1C2, (C2×C6).234(C2×D4), (C2×C3⋊C8).19C22, (S3×C2×C4).13C22, (C3×D4⋊C4)⋊13C2, (C3×C4⋊C4).22C22, (C2×C4).328(C22×S3), SmallGroup(192,340)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊3D12
G = < a,b,c,d | a4=b2=c12=d2=1, bab=cac-1=dad=a-1, cbc-1=a-1b, dbd=ab, dcd=c-1 >
Subgroups: 552 in 162 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C22×S3, C22×S3, C22×C6, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C24⋊C2, C2×C3⋊C8, D6⋊C4, D4⋊S3, C3×C4⋊C4, C2×C24, C2×Dic6, S3×C2×C4, C2×D12, C2×D12, D4⋊2S3, C22×Dic3, C2×C3⋊D4, C6×D4, D4⋊D4, C6.SD16, D6⋊C8, C3×D4⋊C4, C12⋊D4, C2×C24⋊C2, C2×D4⋊S3, C2×D4⋊2S3, D4⋊3D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C22≀C2, C4○D8, C8⋊C22, C2×D12, S3×D4, D4⋊D4, D6⋊D4, D8⋊S3, Q8.7D6, D4⋊3D12
►On 96 points
(1 40 33 68)(2 69 34 41)(3 42 35 70)(4 71 36 43)(5 44 25 72)(6 61 26 45)(7 46 27 62)(8 63 28 47)(9 48 29 64)(10 65 30 37)(11 38 31 66)(12 67 32 39)(13 73 57 86)(14 87 58 74)(15 75 59 88)(16 89 60 76)(17 77 49 90)(18 91 50 78)(19 79 51 92)(20 93 52 80)(21 81 53 94)(22 95 54 82)(23 83 55 96)(24 85 56 84)
(1 57)(2 87)(3 59)(4 89)(5 49)(6 91)(7 51)(8 93)(9 53)(10 95)(11 55)(12 85)(13 33)(14 69)(15 35)(16 71)(17 25)(18 61)(19 27)(20 63)(21 29)(22 65)(23 31)(24 67)(26 78)(28 80)(30 82)(32 84)(34 74)(36 76)(37 54)(38 83)(39 56)(40 73)(41 58)(42 75)(43 60)(44 77)(45 50)(46 79)(47 52)(48 81)(62 92)(64 94)(66 96)(68 86)(70 88)(72 90)
(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 88)(14 87)(15 86)(16 85)(17 96)(18 95)(19 94)(20 93)(21 92)(22 91)(23 90)(24 89)(25 31)(26 30)(27 29)(32 36)(33 35)(37 61)(38 72)(39 71)(40 70)(41 69)(42 68)(43 67)(44 66)(45 65)(46 64)(47 63)(48 62)(49 83)(50 82)(51 81)(52 80)(53 79)(54 78)(55 77)(56 76)(57 75)(58 74)(59 73)(60 84)
G:=sub<Sym(96)| (1,40,33,68)(2,69,34,41)(3,42,35,70)(4,71,36,43)(5,44,25,72)(6,61,26,45)(7,46,27,62)(8,63,28,47)(9,48,29,64)(10,65,30,37)(11,38,31,66)(12,67,32,39)(13,73,57,86)(14,87,58,74)(15,75,59,88)(16,89,60,76)(17,77,49,90)(18,91,50,78)(19,79,51,92)(20,93,52,80)(21,81,53,94)(22,95,54,82)(23,83,55,96)(24,85,56,84), (1,57)(2,87)(3,59)(4,89)(5,49)(6,91)(7,51)(8,93)(9,53)(10,95)(11,55)(12,85)(13,33)(14,69)(15,35)(16,71)(17,25)(18,61)(19,27)(20,63)(21,29)(22,65)(23,31)(24,67)(26,78)(28,80)(30,82)(32,84)(34,74)(36,76)(37,54)(38,83)(39,56)(40,73)(41,58)(42,75)(43,60)(44,77)(45,50)(46,79)(47,52)(48,81)(62,92)(64,94)(66,96)(68,86)(70,88)(72,90), (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)(13,88)(14,87)(15,86)(16,85)(17,96)(18,95)(19,94)(20,93)(21,92)(22,91)(23,90)(24,89)(25,31)(26,30)(27,29)(32,36)(33,35)(37,61)(38,72)(39,71)(40,70)(41,69)(42,68)(43,67)(44,66)(45,65)(46,64)(47,63)(48,62)(49,83)(50,82)(51,81)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,84)>;
G:=Group( (1,40,33,68)(2,69,34,41)(3,42,35,70)(4,71,36,43)(5,44,25,72)(6,61,26,45)(7,46,27,62)(8,63,28,47)(9,48,29,64)(10,65,30,37)(11,38,31,66)(12,67,32,39)(13,73,57,86)(14,87,58,74)(15,75,59,88)(16,89,60,76)(17,77,49,90)(18,91,50,78)(19,79,51,92)(20,93,52,80)(21,81,53,94)(22,95,54,82)(23,83,55,96)(24,85,56,84), (1,57)(2,87)(3,59)(4,89)(5,49)(6,91)(7,51)(8,93)(9,53)(10,95)(11,55)(12,85)(13,33)(14,69)(15,35)(16,71)(17,25)(18,61)(19,27)(20,63)(21,29)(22,65)(23,31)(24,67)(26,78)(28,80)(30,82)(32,84)(34,74)(36,76)(37,54)(38,83)(39,56)(40,73)(41,58)(42,75)(43,60)(44,77)(45,50)(46,79)(47,52)(48,81)(62,92)(64,94)(66,96)(68,86)(70,88)(72,90), (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)(13,88)(14,87)(15,86)(16,85)(17,96)(18,95)(19,94)(20,93)(21,92)(22,91)(23,90)(24,89)(25,31)(26,30)(27,29)(32,36)(33,35)(37,61)(38,72)(39,71)(40,70)(41,69)(42,68)(43,67)(44,66)(45,65)(46,64)(47,63)(48,62)(49,83)(50,82)(51,81)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,84) );
G=PermutationGroup([[(1,40,33,68),(2,69,34,41),(3,42,35,70),(4,71,36,43),(5,44,25,72),(6,61,26,45),(7,46,27,62),(8,63,28,47),(9,48,29,64),(10,65,30,37),(11,38,31,66),(12,67,32,39),(13,73,57,86),(14,87,58,74),(15,75,59,88),(16,89,60,76),(17,77,49,90),(18,91,50,78),(19,79,51,92),(20,93,52,80),(21,81,53,94),(22,95,54,82),(23,83,55,96),(24,85,56,84)], [(1,57),(2,87),(3,59),(4,89),(5,49),(6,91),(7,51),(8,93),(9,53),(10,95),(11,55),(12,85),(13,33),(14,69),(15,35),(16,71),(17,25),(18,61),(19,27),(20,63),(21,29),(22,65),(23,31),(24,67),(26,78),(28,80),(30,82),(32,84),(34,74),(36,76),(37,54),(38,83),(39,56),(40,73),(41,58),(42,75),(43,60),(44,77),(45,50),(46,79),(47,52),(48,81),(62,92),(64,94),(66,96),(68,86),(70,88),(72,90)], [(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),(13,88),(14,87),(15,86),(16,85),(17,96),(18,95),(19,94),(20,93),(21,92),(22,91),(23,90),(24,89),(25,31),(26,30),(27,29),(32,36),(33,35),(37,61),(38,72),(39,71),(40,70),(41,69),(42,68),(43,67),(44,66),(45,65),(46,64),(47,63),(48,62),(49,83),(50,82),(51,81),(52,80),(53,79),(54,78),(55,77),(56,76),(57,75),(58,74),(59,73),(60,84)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 12 | 24 | 2 | 2 | 2 | 6 | 6 | 8 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 4 | 4 | 12 | 12 | 4 | 4 | 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 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D4 | D6 | D6 | D6 | D12 | C4○D8 | C8⋊C22 | S3×D4 | S3×D4 | D8⋊S3 | Q8.7D6 |
| kernel | D4⋊3D12 | C6.SD16 | D6⋊C8 | C3×D4⋊C4 | C12⋊D4 | C2×C24⋊C2 | C2×D4⋊S3 | C2×D4⋊2S3 | D4⋊C4 | Dic6 | C2×Dic3 | C3×D4 | C22×S3 | C4⋊C4 | C2×C8 | C2×D4 | D4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of D4⋊3D12 ►in GL4(𝔽73) generated by
| 72 | 71 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 32 | 32 | 0 | 0 |
| 57 | 41 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 46 | 19 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 14 | 66 |
| 0 | 0 | 7 | 7 |
| 72 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(73))| [72,1,0,0,71,1,0,0,0,0,1,0,0,0,0,1],[32,57,0,0,32,41,0,0,0,0,72,0,0,0,0,72],[46,0,0,0,19,27,0,0,0,0,14,7,0,0,66,7],[72,1,0,0,0,1,0,0,0,0,72,0,0,0,1,1] >;
D4⋊3D12 in GAP, Magma, Sage, TeX
D_4\rtimes_3D_{12} % in TeX
G:=Group("D4:3D12"); // GroupNames label
G:=SmallGroup(192,340);
// by ID
G=gap.SmallGroup(192,340);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,254,219,58,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^12=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^-1*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations