metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊2D4, Dic3⋊4SD16, C4.82(S3×D4), D4⋊C4⋊9S3, C4⋊C4.131D6, C3⋊1(C4⋊SD16), (C2×D4).18D6, C6.D8⋊2C2, (C2×C8).113D6, Dic3⋊C8⋊10C2, C4.1(C4○D12), C12.2(C4○D4), C12⋊3D4.3C2, C12.101(C2×D4), C2.9(D8⋊S3), C2.10(S3×SD16), C6.21(C2×SD16), Dic6⋊C4⋊3C2, C6.13(C4⋊D4), C6.26(C8⋊C22), (C6×D4).23C22, C22.164(S3×D4), C2.16(Dic3⋊D4), (C2×C24).124C22, (C2×C12).202C23, (C2×Dic3).136D4, (C2×D12).45C22, (C4×Dic3).6C22, (C2×Dic6).51C22, (C2×D4.S3)⋊2C2, (C2×C24⋊C2)⋊13C2, (C2×C3⋊C8).8C22, (C3×D4⋊C4)⋊9C2, (C2×C6).215(C2×D4), (C3×C4⋊C4).7C22, (C2×C4).309(C22×S3), SmallGroup(192,321)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6⋊2D4
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=dad=a-1, cac-1=a5, bc=cb, dbd=a3b, dcd=c-1 >
Subgroups: 440 in 128 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C2×D4, C2×D4, C2×Q8, C3⋊C8, C24, Dic6, Dic6, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, D4⋊C4, D4⋊C4, C4⋊C8, C4×Q8, C4⋊1D4, C2×SD16, C24⋊C2, C2×C3⋊C8, C4×Dic3, C4×Dic3, Dic3⋊C4, D4.S3, C3×C4⋊C4, C2×C24, C2×Dic6, C2×D12, C2×C3⋊D4, C6×D4, C4⋊SD16, C6.D8, Dic3⋊C8, C3×D4⋊C4, Dic6⋊C4, C2×C24⋊C2, C2×D4.S3, C12⋊3D4, Dic6⋊2D4
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C4○D4, C22×S3, C4⋊D4, C2×SD16, C8⋊C22, C4○D12, S3×D4, C4⋊SD16, Dic3⋊D4, D8⋊S3, S3×SD16, Dic6⋊2D4
►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 54 7 60)(2 53 8 59)(3 52 9 58)(4 51 10 57)(5 50 11 56)(6 49 12 55)(13 75 19 81)(14 74 20 80)(15 73 21 79)(16 84 22 78)(17 83 23 77)(18 82 24 76)(25 43 31 37)(26 42 32 48)(27 41 33 47)(28 40 34 46)(29 39 35 45)(30 38 36 44)(61 87 67 93)(62 86 68 92)(63 85 69 91)(64 96 70 90)(65 95 71 89)(66 94 72 88)
(1 88 35 17)(2 93 36 22)(3 86 25 15)(4 91 26 20)(5 96 27 13)(6 89 28 18)(7 94 29 23)(8 87 30 16)(9 92 31 21)(10 85 32 14)(11 90 33 19)(12 95 34 24)(37 79 58 62)(38 84 59 67)(39 77 60 72)(40 82 49 65)(41 75 50 70)(42 80 51 63)(43 73 52 68)(44 78 53 61)(45 83 54 66)(46 76 55 71)(47 81 56 64)(48 74 57 69)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 92)(14 91)(15 90)(16 89)(17 88)(18 87)(19 86)(20 85)(21 96)(22 95)(23 94)(24 93)(25 33)(26 32)(27 31)(28 30)(34 36)(37 38)(39 48)(40 47)(41 46)(42 45)(43 44)(49 56)(50 55)(51 54)(52 53)(57 60)(58 59)(61 73)(62 84)(63 83)(64 82)(65 81)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)
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), (1,54,7,60)(2,53,8,59)(3,52,9,58)(4,51,10,57)(5,50,11,56)(6,49,12,55)(13,75,19,81)(14,74,20,80)(15,73,21,79)(16,84,22,78)(17,83,23,77)(18,82,24,76)(25,43,31,37)(26,42,32,48)(27,41,33,47)(28,40,34,46)(29,39,35,45)(30,38,36,44)(61,87,67,93)(62,86,68,92)(63,85,69,91)(64,96,70,90)(65,95,71,89)(66,94,72,88), (1,88,35,17)(2,93,36,22)(3,86,25,15)(4,91,26,20)(5,96,27,13)(6,89,28,18)(7,94,29,23)(8,87,30,16)(9,92,31,21)(10,85,32,14)(11,90,33,19)(12,95,34,24)(37,79,58,62)(38,84,59,67)(39,77,60,72)(40,82,49,65)(41,75,50,70)(42,80,51,63)(43,73,52,68)(44,78,53,61)(45,83,54,66)(46,76,55,71)(47,81,56,64)(48,74,57,69), (2,12)(3,11)(4,10)(5,9)(6,8)(13,92)(14,91)(15,90)(16,89)(17,88)(18,87)(19,86)(20,85)(21,96)(22,95)(23,94)(24,93)(25,33)(26,32)(27,31)(28,30)(34,36)(37,38)(39,48)(40,47)(41,46)(42,45)(43,44)(49,56)(50,55)(51,54)(52,53)(57,60)(58,59)(61,73)(62,84)(63,83)(64,82)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)>;
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), (1,54,7,60)(2,53,8,59)(3,52,9,58)(4,51,10,57)(5,50,11,56)(6,49,12,55)(13,75,19,81)(14,74,20,80)(15,73,21,79)(16,84,22,78)(17,83,23,77)(18,82,24,76)(25,43,31,37)(26,42,32,48)(27,41,33,47)(28,40,34,46)(29,39,35,45)(30,38,36,44)(61,87,67,93)(62,86,68,92)(63,85,69,91)(64,96,70,90)(65,95,71,89)(66,94,72,88), (1,88,35,17)(2,93,36,22)(3,86,25,15)(4,91,26,20)(5,96,27,13)(6,89,28,18)(7,94,29,23)(8,87,30,16)(9,92,31,21)(10,85,32,14)(11,90,33,19)(12,95,34,24)(37,79,58,62)(38,84,59,67)(39,77,60,72)(40,82,49,65)(41,75,50,70)(42,80,51,63)(43,73,52,68)(44,78,53,61)(45,83,54,66)(46,76,55,71)(47,81,56,64)(48,74,57,69), (2,12)(3,11)(4,10)(5,9)(6,8)(13,92)(14,91)(15,90)(16,89)(17,88)(18,87)(19,86)(20,85)(21,96)(22,95)(23,94)(24,93)(25,33)(26,32)(27,31)(28,30)(34,36)(37,38)(39,48)(40,47)(41,46)(42,45)(43,44)(49,56)(50,55)(51,54)(52,53)(57,60)(58,59)(61,73)(62,84)(63,83)(64,82)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74) );
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)], [(1,54,7,60),(2,53,8,59),(3,52,9,58),(4,51,10,57),(5,50,11,56),(6,49,12,55),(13,75,19,81),(14,74,20,80),(15,73,21,79),(16,84,22,78),(17,83,23,77),(18,82,24,76),(25,43,31,37),(26,42,32,48),(27,41,33,47),(28,40,34,46),(29,39,35,45),(30,38,36,44),(61,87,67,93),(62,86,68,92),(63,85,69,91),(64,96,70,90),(65,95,71,89),(66,94,72,88)], [(1,88,35,17),(2,93,36,22),(3,86,25,15),(4,91,26,20),(5,96,27,13),(6,89,28,18),(7,94,29,23),(8,87,30,16),(9,92,31,21),(10,85,32,14),(11,90,33,19),(12,95,34,24),(37,79,58,62),(38,84,59,67),(39,77,60,72),(40,82,49,65),(41,75,50,70),(42,80,51,63),(43,73,52,68),(44,78,53,61),(45,83,54,66),(46,76,55,71),(47,81,56,64),(48,74,57,69)], [(2,12),(3,11),(4,10),(5,9),(6,8),(13,92),(14,91),(15,90),(16,89),(17,88),(18,87),(19,86),(20,85),(21,96),(22,95),(23,94),(24,93),(25,33),(26,32),(27,31),(28,30),(34,36),(37,38),(39,48),(40,47),(41,46),(42,45),(43,44),(49,56),(50,55),(51,54),(52,53),(57,60),(58,59),(61,73),(62,84),(63,83),(64,82),(65,81),(66,80),(67,79),(68,78),(69,77),(70,76),(71,75),(72,74)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 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 | 24 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 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 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | SD16 | C4○D4 | C4○D12 | C8⋊C22 | S3×D4 | S3×D4 | D8⋊S3 | S3×SD16 |
| kernel | Dic6⋊2D4 | C6.D8 | Dic3⋊C8 | C3×D4⋊C4 | Dic6⋊C4 | C2×C24⋊C2 | C2×D4.S3 | C12⋊3D4 | D4⋊C4 | Dic6 | C2×Dic3 | C4⋊C4 | C2×C8 | C2×D4 | Dic3 | C12 | C4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of Dic6⋊2D4 ►in GL4(𝔽73) generated by
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 |
| 0 | 0 | 1 | 0 |
| 67 | 67 | 0 | 0 |
| 67 | 6 | 0 | 0 |
| 0 | 0 | 14 | 66 |
| 0 | 0 | 7 | 59 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 43 |
| 0 | 0 | 30 | 13 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(73))| [0,72,0,0,1,0,0,0,0,0,1,1,0,0,72,0],[67,67,0,0,67,6,0,0,0,0,14,7,0,0,66,59],[1,0,0,0,0,1,0,0,0,0,60,30,0,0,43,13],[1,0,0,0,0,72,0,0,0,0,0,1,0,0,1,0] >;
Dic6⋊2D4 in GAP, Magma, Sage, TeX
{\rm Dic}_6\rtimes_2D_4 % in TeX
G:=Group("Dic6:2D4"); // GroupNames label
G:=SmallGroup(192,321);
// by ID
G=gap.SmallGroup(192,321);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,135,268,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^4=d^2=1,b^2=a^6,b*a*b^-1=d*a*d=a^-1,c*a*c^-1=a^5,b*c=c*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations