metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊14D4, Dic6⋊13D4, C23.17D12, (C2×C8).3D6, (C2×D24)⋊3C2, C22⋊C8⋊6S3, C3⋊1(D4⋊D4), C6.9(C4○D8), C4.122(S3×D4), C2.D24⋊9C2, C12⋊7D4⋊16C2, C6.10C22≀C2, (C2×C12).241D4, (C2×C4).119D12, C12.334(C2×D4), C2.Dic12⋊6C2, (C22×C6).54D4, C2.11(C4○D24), C2.13(C8⋊D6), C6.10(C8⋊C22), (C22×C4).100D6, C2.13(D6⋊D4), (C2×C24).119C22, (C2×C12).744C23, C22.107(C2×D12), C4⋊Dic3.13C22, (C2×D12).193C22, (C22×C12).97C22, (C2×Dic6).211C22, (C2×C4○D12)⋊1C2, (C3×C22⋊C8)⋊8C2, (C2×C24⋊C2)⋊11C2, (C2×C6).127(C2×D4), (C2×C4).689(C22×S3), SmallGroup(192,293)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊14D4
G = < a,b,c,d | a12=b2=c4=d2=1, bab=cac-1=a-1, ad=da, cbc-1=ab, dbd=a6b, dcd=c-1 >
Subgroups: 576 in 162 conjugacy classes, 43 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C24⋊C2, D24, C4⋊Dic3, D6⋊C4, C2×C24, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, D4⋊D4, C2.Dic12, C2.D24, C3×C22⋊C8, C2×C24⋊C2, C2×D24, C12⋊7D4, C2×C4○D12, D12⋊14D4
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, C4○D24, C8⋊D6, D12⋊14D4
►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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 45)(14 44)(15 43)(16 42)(17 41)(18 40)(19 39)(20 38)(21 37)(22 48)(23 47)(24 46)(49 79)(50 78)(51 77)(52 76)(53 75)(54 74)(55 73)(56 84)(57 83)(58 82)(59 81)(60 80)(61 92)(62 91)(63 90)(64 89)(65 88)(66 87)(67 86)(68 85)(69 96)(70 95)(71 94)(72 93)
(1 66 16 58)(2 65 17 57)(3 64 18 56)(4 63 19 55)(5 62 20 54)(6 61 21 53)(7 72 22 52)(8 71 23 51)(9 70 24 50)(10 69 13 49)(11 68 14 60)(12 67 15 59)(25 91 37 74)(26 90 38 73)(27 89 39 84)(28 88 40 83)(29 87 41 82)(30 86 42 81)(31 85 43 80)(32 96 44 79)(33 95 45 78)(34 94 46 77)(35 93 47 76)(36 92 48 75)
(1 58)(2 59)(3 60)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 71)(22 72)(23 61)(24 62)(25 83)(26 84)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 79)(34 80)(35 81)(36 82)(37 88)(38 89)(39 90)(40 91)(41 92)(42 93)(43 94)(44 95)(45 96)(46 85)(47 86)(48 87)
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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,45)(14,44)(15,43)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,48)(23,47)(24,46)(49,79)(50,78)(51,77)(52,76)(53,75)(54,74)(55,73)(56,84)(57,83)(58,82)(59,81)(60,80)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,96)(70,95)(71,94)(72,93), (1,66,16,58)(2,65,17,57)(3,64,18,56)(4,63,19,55)(5,62,20,54)(6,61,21,53)(7,72,22,52)(8,71,23,51)(9,70,24,50)(10,69,13,49)(11,68,14,60)(12,67,15,59)(25,91,37,74)(26,90,38,73)(27,89,39,84)(28,88,40,83)(29,87,41,82)(30,86,42,81)(31,85,43,80)(32,96,44,79)(33,95,45,78)(34,94,46,77)(35,93,47,76)(36,92,48,75), (1,58)(2,59)(3,60)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,61)(24,62)(25,83)(26,84)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,81)(36,82)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,94)(44,95)(45,96)(46,85)(47,86)(48,87)>;
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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,45)(14,44)(15,43)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,48)(23,47)(24,46)(49,79)(50,78)(51,77)(52,76)(53,75)(54,74)(55,73)(56,84)(57,83)(58,82)(59,81)(60,80)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,96)(70,95)(71,94)(72,93), (1,66,16,58)(2,65,17,57)(3,64,18,56)(4,63,19,55)(5,62,20,54)(6,61,21,53)(7,72,22,52)(8,71,23,51)(9,70,24,50)(10,69,13,49)(11,68,14,60)(12,67,15,59)(25,91,37,74)(26,90,38,73)(27,89,39,84)(28,88,40,83)(29,87,41,82)(30,86,42,81)(31,85,43,80)(32,96,44,79)(33,95,45,78)(34,94,46,77)(35,93,47,76)(36,92,48,75), (1,58)(2,59)(3,60)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,61)(24,62)(25,83)(26,84)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,81)(36,82)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,94)(44,95)(45,96)(46,85)(47,86)(48,87) );
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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,36),(8,35),(9,34),(10,33),(11,32),(12,31),(13,45),(14,44),(15,43),(16,42),(17,41),(18,40),(19,39),(20,38),(21,37),(22,48),(23,47),(24,46),(49,79),(50,78),(51,77),(52,76),(53,75),(54,74),(55,73),(56,84),(57,83),(58,82),(59,81),(60,80),(61,92),(62,91),(63,90),(64,89),(65,88),(66,87),(67,86),(68,85),(69,96),(70,95),(71,94),(72,93)], [(1,66,16,58),(2,65,17,57),(3,64,18,56),(4,63,19,55),(5,62,20,54),(6,61,21,53),(7,72,22,52),(8,71,23,51),(9,70,24,50),(10,69,13,49),(11,68,14,60),(12,67,15,59),(25,91,37,74),(26,90,38,73),(27,89,39,84),(28,88,40,83),(29,87,41,82),(30,86,42,81),(31,85,43,80),(32,96,44,79),(33,95,45,78),(34,94,46,77),(35,93,47,76),(36,92,48,75)], [(1,58),(2,59),(3,60),(4,49),(5,50),(6,51),(7,52),(8,53),(9,54),(10,55),(11,56),(12,57),(13,63),(14,64),(15,65),(16,66),(17,67),(18,68),(19,69),(20,70),(21,71),(22,72),(23,61),(24,62),(25,83),(26,84),(27,73),(28,74),(29,75),(30,76),(31,77),(32,78),(33,79),(34,80),(35,81),(36,82),(37,88),(38,89),(39,90),(40,91),(41,92),(42,93),(43,94),(44,95),(45,96),(46,85),(47,86),(48,87)]])
39 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 | 12E | 12F | 24A | ··· | 24H |
| 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 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 12 | 12 | 24 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D4 | D6 | D6 | D12 | D12 | C4○D8 | C4○D24 | C8⋊C22 | S3×D4 | C8⋊D6 |
| kernel | D12⋊14D4 | C2.Dic12 | C2.D24 | C3×C22⋊C8 | C2×C24⋊C2 | C2×D24 | C12⋊7D4 | C2×C4○D12 | C22⋊C8 | Dic6 | D12 | C2×C12 | C22×C6 | C2×C8 | C22×C4 | C2×C4 | C23 | C6 | C2 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 8 | 1 | 2 | 2 |
Matrix representation of D12⋊14D4 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 14 | 66 |
| 0 | 0 | 7 | 7 |
| 0 | 72 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 18 | 50 |
| 0 | 0 | 68 | 55 |
| 0 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 13 | 30 |
| 0 | 0 | 43 | 60 |
| 0 | 72 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 43 | 60 |
| 0 | 0 | 13 | 30 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,14,7,0,0,66,7],[0,72,0,0,72,0,0,0,0,0,18,68,0,0,50,55],[0,1,0,0,72,0,0,0,0,0,13,43,0,0,30,60],[0,72,0,0,72,0,0,0,0,0,43,13,0,0,60,30] >;
D12⋊14D4 in GAP, Magma, Sage, TeX
D_{12}\rtimes_{14}D_4 % in TeX
G:=Group("D12:14D4"); // GroupNames label
G:=SmallGroup(192,293);
// by ID
G=gap.SmallGroup(192,293);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,226,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a*b,d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations