metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊8D4, C12⋊2SD16, C42.39D6, C4⋊C8⋊10S3, C4⋊2(C24⋊C2), C3⋊2(C4⋊SD16), C4.133(S3×D4), (C2×C8).133D6, (C4×Dic6)⋊18C2, C4⋊D12.5C2, C12.342(C2×D4), (C2×C12).124D4, (C2×C4).135D12, C2.D24⋊13C2, C6.12(C2×SD16), C6.41(C4⋊D4), C2.20(C8⋊D6), C6.17(C8⋊C22), (C4×C12).74C22, C12.331(C4○D4), C2.14(C12⋊D4), (C2×C24).140C22, (C2×C12).758C23, C4.47(Q8⋊3S3), (C2×D12).17C22, C22.121(C2×D12), C4⋊Dic3.276C22, (C2×Dic6).215C22, (C3×C4⋊C8)⋊12C2, (C2×C24⋊C2)⋊20C2, C2.15(C2×C24⋊C2), (C2×C6).141(C2×D4), (C2×C4).703(C22×S3), SmallGroup(192,407)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6⋊8D4
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a9b, dcd=c-1 >
Subgroups: 488 in 128 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, C12, D6, C2×C6, C42, C42, C4⋊C4, C2×C8, SD16, C2×D4, C2×Q8, C24, Dic6, Dic6, D12, C2×Dic3, C2×C12, C22×S3, D4⋊C4, C4⋊C8, C4×Q8, C4⋊1D4, C2×SD16, C24⋊C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×C24, C2×Dic6, C2×D12, C2×D12, C4⋊SD16, C2.D24, C3×C4⋊C8, C4×Dic6, C4⋊D12, C2×C24⋊C2, Dic6⋊8D4
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C4○D4, D12, C22×S3, C4⋊D4, C2×SD16, C8⋊C22, C24⋊C2, C2×D12, S3×D4, Q8⋊3S3, C4⋊SD16, C12⋊D4, C2×C24⋊C2, C8⋊D6, Dic6⋊8D4
►On 96 points
Generators in S96
(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 20 7 14)(2 19 8 13)(3 18 9 24)(4 17 10 23)(5 16 11 22)(6 15 12 21)(25 82 31 76)(26 81 32 75)(27 80 33 74)(28 79 34 73)(29 78 35 84)(30 77 36 83)(37 93 43 87)(38 92 44 86)(39 91 45 85)(40 90 46 96)(41 89 47 95)(42 88 48 94)(49 64 55 70)(50 63 56 69)(51 62 57 68)(52 61 58 67)(53 72 59 66)(54 71 60 65)
(1 32 40 67)(2 33 41 68)(3 34 42 69)(4 35 43 70)(5 36 44 71)(6 25 45 72)(7 26 46 61)(8 27 47 62)(9 28 48 63)(10 29 37 64)(11 30 38 65)(12 31 39 66)(13 80 95 57)(14 81 96 58)(15 82 85 59)(16 83 86 60)(17 84 87 49)(18 73 88 50)(19 74 89 51)(20 75 90 52)(21 76 91 53)(22 77 92 54)(23 78 93 55)(24 79 94 56)
(1 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 36)(10 35)(11 34)(12 33)(13 73)(14 84)(15 83)(16 82)(17 81)(18 80)(19 79)(20 78)(21 77)(22 76)(23 75)(24 74)(37 70)(38 69)(39 68)(40 67)(41 66)(42 65)(43 64)(44 63)(45 62)(46 61)(47 72)(48 71)(49 96)(50 95)(51 94)(52 93)(53 92)(54 91)(55 90)(56 89)(57 88)(58 87)(59 86)(60 85)
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), (1,20,7,14)(2,19,8,13)(3,18,9,24)(4,17,10,23)(5,16,11,22)(6,15,12,21)(25,82,31,76)(26,81,32,75)(27,80,33,74)(28,79,34,73)(29,78,35,84)(30,77,36,83)(37,93,43,87)(38,92,44,86)(39,91,45,85)(40,90,46,96)(41,89,47,95)(42,88,48,94)(49,64,55,70)(50,63,56,69)(51,62,57,68)(52,61,58,67)(53,72,59,66)(54,71,60,65), (1,32,40,67)(2,33,41,68)(3,34,42,69)(4,35,43,70)(5,36,44,71)(6,25,45,72)(7,26,46,61)(8,27,47,62)(9,28,48,63)(10,29,37,64)(11,30,38,65)(12,31,39,66)(13,80,95,57)(14,81,96,58)(15,82,85,59)(16,83,86,60)(17,84,87,49)(18,73,88,50)(19,74,89,51)(20,75,90,52)(21,76,91,53)(22,77,92,54)(23,78,93,55)(24,79,94,56), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,36)(10,35)(11,34)(12,33)(13,73)(14,84)(15,83)(16,82)(17,81)(18,80)(19,79)(20,78)(21,77)(22,76)(23,75)(24,74)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,72)(48,71)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)>;
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), (1,20,7,14)(2,19,8,13)(3,18,9,24)(4,17,10,23)(5,16,11,22)(6,15,12,21)(25,82,31,76)(26,81,32,75)(27,80,33,74)(28,79,34,73)(29,78,35,84)(30,77,36,83)(37,93,43,87)(38,92,44,86)(39,91,45,85)(40,90,46,96)(41,89,47,95)(42,88,48,94)(49,64,55,70)(50,63,56,69)(51,62,57,68)(52,61,58,67)(53,72,59,66)(54,71,60,65), (1,32,40,67)(2,33,41,68)(3,34,42,69)(4,35,43,70)(5,36,44,71)(6,25,45,72)(7,26,46,61)(8,27,47,62)(9,28,48,63)(10,29,37,64)(11,30,38,65)(12,31,39,66)(13,80,95,57)(14,81,96,58)(15,82,85,59)(16,83,86,60)(17,84,87,49)(18,73,88,50)(19,74,89,51)(20,75,90,52)(21,76,91,53)(22,77,92,54)(23,78,93,55)(24,79,94,56), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,36)(10,35)(11,34)(12,33)(13,73)(14,84)(15,83)(16,82)(17,81)(18,80)(19,79)(20,78)(21,77)(22,76)(23,75)(24,74)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,72)(48,71)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85) );
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)], [(1,20,7,14),(2,19,8,13),(3,18,9,24),(4,17,10,23),(5,16,11,22),(6,15,12,21),(25,82,31,76),(26,81,32,75),(27,80,33,74),(28,79,34,73),(29,78,35,84),(30,77,36,83),(37,93,43,87),(38,92,44,86),(39,91,45,85),(40,90,46,96),(41,89,47,95),(42,88,48,94),(49,64,55,70),(50,63,56,69),(51,62,57,68),(52,61,58,67),(53,72,59,66),(54,71,60,65)], [(1,32,40,67),(2,33,41,68),(3,34,42,69),(4,35,43,70),(5,36,44,71),(6,25,45,72),(7,26,46,61),(8,27,47,62),(9,28,48,63),(10,29,37,64),(11,30,38,65),(12,31,39,66),(13,80,95,57),(14,81,96,58),(15,82,85,59),(16,83,86,60),(17,84,87,49),(18,73,88,50),(19,74,89,51),(20,75,90,52),(21,76,91,53),(22,77,92,54),(23,78,93,55),(24,79,94,56)], [(1,32),(2,31),(3,30),(4,29),(5,28),(6,27),(7,26),(8,25),(9,36),(10,35),(11,34),(12,33),(13,73),(14,84),(15,83),(16,82),(17,81),(18,80),(19,79),(20,78),(21,77),(22,76),(23,75),(24,74),(37,70),(38,69),(39,68),(40,67),(41,66),(42,65),(43,64),(44,63),(45,62),(46,61),(47,72),(48,71),(49,96),(50,95),(51,94),(52,93),(53,92),(54,91),(55,90),(56,89),(57,88),(58,87),(59,86),(60,85)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 24 | 24 | 2 | 2 | 2 | 2 | 2 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | SD16 | C4○D4 | D12 | C24⋊C2 | C8⋊C22 | S3×D4 | Q8⋊3S3 | C8⋊D6 |
| kernel | Dic6⋊8D4 | C2.D24 | C3×C4⋊C8 | C4×Dic6 | C4⋊D12 | C2×C24⋊C2 | C4⋊C8 | Dic6 | C2×C12 | C42 | C2×C8 | C12 | C12 | C2×C4 | C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 8 | 1 | 1 | 1 | 2 |
Matrix representation of Dic6⋊8D4 ►in GL4(𝔽73) generated by
| 59 | 66 | 0 | 0 |
| 7 | 66 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 25 | 62 | 0 | 0 |
| 37 | 48 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 44 | 71 |
| 0 | 0 | 56 | 29 |
| 66 | 59 | 0 | 0 |
| 66 | 7 | 0 | 0 |
| 0 | 0 | 29 | 2 |
| 0 | 0 | 18 | 44 |
G:=sub<GL(4,GF(73))| [59,7,0,0,66,66,0,0,0,0,1,0,0,0,0,1],[25,37,0,0,62,48,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,44,56,0,0,71,29],[66,66,0,0,59,7,0,0,0,0,29,18,0,0,2,44] >;
Dic6⋊8D4 in GAP, Magma, Sage, TeX
{\rm Dic}_6\rtimes_8D_4 % in TeX
G:=Group("Dic6:8D4"); // GroupNames label
G:=SmallGroup(192,407);
// by ID
G=gap.SmallGroup(192,407);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,254,219,58,1123,136,6278]);
// 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,a*c=c*a,b*c=c*b,d*b*d=a^9*b,d*c*d=c^-1>;
// generators/relations