metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊7D4, (C3×Q8)⋊6D4, D6⋊C8⋊34C2, C4.64(S3×D4), D6⋊3D4⋊7C2, C3⋊6(D4⋊D4), Q8⋊4(C3⋊D4), (C2×D4).74D6, C12.49(C2×D4), (C2×C8).148D6, C6.59C22≀C2, (C6×SD16)⋊22C2, (C2×SD16)⋊13S3, C6.64(C4○D8), C2.D24⋊36C2, (C2×Q8).141D6, Q8⋊2Dic3⋊30C2, C2.29(Q8⋊3D6), C6.79(C8⋊C22), (C22×S3).37D4, (C6×D4).98C22, C22.269(S3×D4), (C6×Q8).78C22, C2.27(C23⋊2D6), (C2×C12).449C23, (C2×C24).295C22, (C2×Dic3).184D4, C2.30(Q8.7D6), (C2×D12).121C22, C4⋊Dic3.176C22, (C2×D4⋊S3)⋊20C2, C4.44(C2×C3⋊D4), (C2×Q8⋊3S3)⋊2C2, (C2×C6).361(C2×D4), (S3×C2×C4).49C22, (C2×C3⋊C8).158C22, (C2×C4).538(C22×S3), SmallGroup(192,731)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊7D4
G = < a,b,c,d | a12=b2=c4=d2=1, bab=cac-1=a-1, dad=a7, cbc-1=a7b, dbd=a9b, dcd=c-1 >
Subgroups: 536 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, Q8, 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, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C3×Q8, C22×S3, C22×S3, C22×C6, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C2×C3⋊C8, C4⋊Dic3, D4⋊S3, C6.D4, C2×C24, C3×SD16, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, Q8⋊3S3, C2×C3⋊D4, C6×D4, C6×Q8, D4⋊D4, D6⋊C8, C2.D24, Q8⋊2Dic3, C2×D4⋊S3, D6⋊3D4, C6×SD16, C2×Q8⋊3S3, D12⋊7D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C22≀C2, C4○D8, C8⋊C22, S3×D4, C2×C3⋊D4, D4⋊D4, Q8⋊3D6, Q8.7D6, C23⋊2D6, D12⋊7D4
►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 19)(14 18)(15 17)(20 24)(21 23)(25 26)(27 36)(28 35)(29 34)(30 33)(31 32)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)(49 56)(50 55)(51 54)(52 53)(57 60)(58 59)(61 69)(62 68)(63 67)(64 66)(70 72)(73 83)(74 82)(75 81)(76 80)(77 79)(85 87)(88 96)(89 95)(90 94)(91 93)
(1 94 32 18)(2 93 33 17)(3 92 34 16)(4 91 35 15)(5 90 36 14)(6 89 25 13)(7 88 26 24)(8 87 27 23)(9 86 28 22)(10 85 29 21)(11 96 30 20)(12 95 31 19)(37 74 53 67)(38 73 54 66)(39 84 55 65)(40 83 56 64)(41 82 57 63)(42 81 58 62)(43 80 59 61)(44 79 60 72)(45 78 49 71)(46 77 50 70)(47 76 51 69)(48 75 52 68)
(1 77)(2 84)(3 79)(4 74)(5 81)(6 76)(7 83)(8 78)(9 73)(10 80)(11 75)(12 82)(13 51)(14 58)(15 53)(16 60)(17 55)(18 50)(19 57)(20 52)(21 59)(22 54)(23 49)(24 56)(25 69)(26 64)(27 71)(28 66)(29 61)(30 68)(31 63)(32 70)(33 65)(34 72)(35 67)(36 62)(37 91)(38 86)(39 93)(40 88)(41 95)(42 90)(43 85)(44 92)(45 87)(46 94)(47 89)(48 96)
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)(13,19)(14,18)(15,17)(20,24)(21,23)(25,26)(27,36)(28,35)(29,34)(30,33)(31,32)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)(49,56)(50,55)(51,54)(52,53)(57,60)(58,59)(61,69)(62,68)(63,67)(64,66)(70,72)(73,83)(74,82)(75,81)(76,80)(77,79)(85,87)(88,96)(89,95)(90,94)(91,93), (1,94,32,18)(2,93,33,17)(3,92,34,16)(4,91,35,15)(5,90,36,14)(6,89,25,13)(7,88,26,24)(8,87,27,23)(9,86,28,22)(10,85,29,21)(11,96,30,20)(12,95,31,19)(37,74,53,67)(38,73,54,66)(39,84,55,65)(40,83,56,64)(41,82,57,63)(42,81,58,62)(43,80,59,61)(44,79,60,72)(45,78,49,71)(46,77,50,70)(47,76,51,69)(48,75,52,68), (1,77)(2,84)(3,79)(4,74)(5,81)(6,76)(7,83)(8,78)(9,73)(10,80)(11,75)(12,82)(13,51)(14,58)(15,53)(16,60)(17,55)(18,50)(19,57)(20,52)(21,59)(22,54)(23,49)(24,56)(25,69)(26,64)(27,71)(28,66)(29,61)(30,68)(31,63)(32,70)(33,65)(34,72)(35,67)(36,62)(37,91)(38,86)(39,93)(40,88)(41,95)(42,90)(43,85)(44,92)(45,87)(46,94)(47,89)(48,96)>;
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)(13,19)(14,18)(15,17)(20,24)(21,23)(25,26)(27,36)(28,35)(29,34)(30,33)(31,32)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)(49,56)(50,55)(51,54)(52,53)(57,60)(58,59)(61,69)(62,68)(63,67)(64,66)(70,72)(73,83)(74,82)(75,81)(76,80)(77,79)(85,87)(88,96)(89,95)(90,94)(91,93), (1,94,32,18)(2,93,33,17)(3,92,34,16)(4,91,35,15)(5,90,36,14)(6,89,25,13)(7,88,26,24)(8,87,27,23)(9,86,28,22)(10,85,29,21)(11,96,30,20)(12,95,31,19)(37,74,53,67)(38,73,54,66)(39,84,55,65)(40,83,56,64)(41,82,57,63)(42,81,58,62)(43,80,59,61)(44,79,60,72)(45,78,49,71)(46,77,50,70)(47,76,51,69)(48,75,52,68), (1,77)(2,84)(3,79)(4,74)(5,81)(6,76)(7,83)(8,78)(9,73)(10,80)(11,75)(12,82)(13,51)(14,58)(15,53)(16,60)(17,55)(18,50)(19,57)(20,52)(21,59)(22,54)(23,49)(24,56)(25,69)(26,64)(27,71)(28,66)(29,61)(30,68)(31,63)(32,70)(33,65)(34,72)(35,67)(36,62)(37,91)(38,86)(39,93)(40,88)(41,95)(42,90)(43,85)(44,92)(45,87)(46,94)(47,89)(48,96) );
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),(13,19),(14,18),(15,17),(20,24),(21,23),(25,26),(27,36),(28,35),(29,34),(30,33),(31,32),(37,48),(38,47),(39,46),(40,45),(41,44),(42,43),(49,56),(50,55),(51,54),(52,53),(57,60),(58,59),(61,69),(62,68),(63,67),(64,66),(70,72),(73,83),(74,82),(75,81),(76,80),(77,79),(85,87),(88,96),(89,95),(90,94),(91,93)], [(1,94,32,18),(2,93,33,17),(3,92,34,16),(4,91,35,15),(5,90,36,14),(6,89,25,13),(7,88,26,24),(8,87,27,23),(9,86,28,22),(10,85,29,21),(11,96,30,20),(12,95,31,19),(37,74,53,67),(38,73,54,66),(39,84,55,65),(40,83,56,64),(41,82,57,63),(42,81,58,62),(43,80,59,61),(44,79,60,72),(45,78,49,71),(46,77,50,70),(47,76,51,69),(48,75,52,68)], [(1,77),(2,84),(3,79),(4,74),(5,81),(6,76),(7,83),(8,78),(9,73),(10,80),(11,75),(12,82),(13,51),(14,58),(15,53),(16,60),(17,55),(18,50),(19,57),(20,52),(21,59),(22,54),(23,49),(24,56),(25,69),(26,64),(27,71),(28,66),(29,61),(30,68),(31,63),(32,70),(33,65),(34,72),(35,67),(36,62),(37,91),(38,86),(39,93),(40,88),(41,95),(42,90),(43,85),(44,92),(45,87),(46,94),(47,89),(48,96)]])
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 | 8 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 24 | 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 | C3⋊D4 | C4○D8 | C8⋊C22 | S3×D4 | S3×D4 | Q8⋊3D6 | Q8.7D6 |
| kernel | D12⋊7D4 | D6⋊C8 | C2.D24 | Q8⋊2Dic3 | C2×D4⋊S3 | D6⋊3D4 | C6×SD16 | C2×Q8⋊3S3 | C2×SD16 | D12 | C2×Dic3 | C3×Q8 | C22×S3 | C2×C8 | C2×D4 | C2×Q8 | Q8 | 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 D12⋊7D4 ►in GL4(𝔽73) generated by
| 1 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 2 |
| 0 | 0 | 72 | 1 |
| 1 | 1 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 2 |
| 0 | 0 | 0 | 1 |
| 43 | 13 | 0 | 0 |
| 43 | 30 | 0 | 0 |
| 0 | 0 | 0 | 61 |
| 0 | 0 | 67 | 0 |
| 43 | 13 | 0 | 0 |
| 60 | 30 | 0 | 0 |
| 0 | 0 | 32 | 41 |
| 0 | 0 | 16 | 41 |
G:=sub<GL(4,GF(73))| [1,72,0,0,1,0,0,0,0,0,72,72,0,0,2,1],[1,0,0,0,1,72,0,0,0,0,72,0,0,0,2,1],[43,43,0,0,13,30,0,0,0,0,0,67,0,0,61,0],[43,60,0,0,13,30,0,0,0,0,32,16,0,0,41,41] >;
D12⋊7D4 in GAP, Magma, Sage, TeX
D_{12}\rtimes_7D_4 % in TeX
G:=Group("D12:7D4"); // GroupNames label
G:=SmallGroup(192,731);
// by ID
G=gap.SmallGroup(192,731);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,758,135,184,570,297,136,6278]);
// 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,d*a*d=a^7,c*b*c^-1=a^7*b,d*b*d=a^9*b,d*c*d=c^-1>;
// generators/relations