metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D24⋊6C4, Dic3⋊5D8, C3⋊3(C4×D8), C8⋊7(C4×S3), C24⋊4(C2×C4), C2.3(S3×D8), D12⋊7(C2×C4), C6.52(C4×D4), C6.26(C2×D8), C2.D8⋊13S3, C4⋊C4.166D6, (C8×Dic3)⋊3C2, (C2×D24).9C2, (C2×C8).225D6, Dic3⋊5D4⋊7C2, C6.73(C4○D8), C6.D8⋊19C2, C22.87(S3×D4), C12.35(C4○D4), C12.46(C22×C4), (C2×C24).77C22, C4.7(Q8⋊3S3), C2.3(D24⋊C2), (C2×C12).288C23, (C2×Dic3).207D4, (C2×D12).80C22, C2.12(Dic3⋊5D4), (C4×Dic3).231C22, C4.43(S3×C2×C4), (C3×C2.D8)⋊2C2, (C2×C6).293(C2×D4), (C3×C4⋊C4).81C22, (C2×C3⋊C8).229C22, (C2×C4).391(C22×S3), SmallGroup(192,431)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3⋊5D8
G = < a,b,c,d | a6=c8=d2=1, b2=a3, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >
Subgroups: 448 in 134 conjugacy classes, 51 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C8, C2×C4, C2×C4, D4, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C3⋊C8, C24, C4×S3, D12, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C4×C8, D4⋊C4, C2.D8, C4×D4, C2×D8, D24, C2×C3⋊C8, C4×Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, S3×C2×C4, C2×D12, C4×D8, C6.D8, C8×Dic3, C3×C2.D8, Dic3⋊5D4, C2×D24, Dic3⋊5D8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, D8, C22×C4, C2×D4, C4○D4, C4×S3, C22×S3, C4×D4, C2×D8, C4○D8, S3×C2×C4, S3×D4, Q8⋊3S3, C4×D8, Dic3⋊5D4, S3×D8, D24⋊C2, Dic3⋊5D8
►On 96 points
(1 88 50 16 29 77)(2 81 51 9 30 78)(3 82 52 10 31 79)(4 83 53 11 32 80)(5 84 54 12 25 73)(6 85 55 13 26 74)(7 86 56 14 27 75)(8 87 49 15 28 76)(17 89 62 39 42 67)(18 90 63 40 43 68)(19 91 64 33 44 69)(20 92 57 34 45 70)(21 93 58 35 46 71)(22 94 59 36 47 72)(23 95 60 37 48 65)(24 96 61 38 41 66)
(1 18 16 40)(2 19 9 33)(3 20 10 34)(4 21 11 35)(5 22 12 36)(6 23 13 37)(7 24 14 38)(8 17 15 39)(25 59 84 72)(26 60 85 65)(27 61 86 66)(28 62 87 67)(29 63 88 68)(30 64 81 69)(31 57 82 70)(32 58 83 71)(41 75 96 56)(42 76 89 49)(43 77 90 50)(44 78 91 51)(45 79 92 52)(46 80 93 53)(47 73 94 54)(48 74 95 55)
(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 5)(2 4)(6 8)(9 11)(12 16)(13 15)(17 23)(18 22)(19 21)(25 50)(26 49)(27 56)(28 55)(29 54)(30 53)(31 52)(32 51)(33 35)(36 40)(37 39)(41 61)(42 60)(43 59)(44 58)(45 57)(46 64)(47 63)(48 62)(65 89)(66 96)(67 95)(68 94)(69 93)(70 92)(71 91)(72 90)(73 88)(74 87)(75 86)(76 85)(77 84)(78 83)(79 82)(80 81)
G:=sub<Sym(96)| (1,88,50,16,29,77)(2,81,51,9,30,78)(3,82,52,10,31,79)(4,83,53,11,32,80)(5,84,54,12,25,73)(6,85,55,13,26,74)(7,86,56,14,27,75)(8,87,49,15,28,76)(17,89,62,39,42,67)(18,90,63,40,43,68)(19,91,64,33,44,69)(20,92,57,34,45,70)(21,93,58,35,46,71)(22,94,59,36,47,72)(23,95,60,37,48,65)(24,96,61,38,41,66), (1,18,16,40)(2,19,9,33)(3,20,10,34)(4,21,11,35)(5,22,12,36)(6,23,13,37)(7,24,14,38)(8,17,15,39)(25,59,84,72)(26,60,85,65)(27,61,86,66)(28,62,87,67)(29,63,88,68)(30,64,81,69)(31,57,82,70)(32,58,83,71)(41,75,96,56)(42,76,89,49)(43,77,90,50)(44,78,91,51)(45,79,92,52)(46,80,93,53)(47,73,94,54)(48,74,95,55), (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), (1,5)(2,4)(6,8)(9,11)(12,16)(13,15)(17,23)(18,22)(19,21)(25,50)(26,49)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,35)(36,40)(37,39)(41,61)(42,60)(43,59)(44,58)(45,57)(46,64)(47,63)(48,62)(65,89)(66,96)(67,95)(68,94)(69,93)(70,92)(71,91)(72,90)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,81)>;
G:=Group( (1,88,50,16,29,77)(2,81,51,9,30,78)(3,82,52,10,31,79)(4,83,53,11,32,80)(5,84,54,12,25,73)(6,85,55,13,26,74)(7,86,56,14,27,75)(8,87,49,15,28,76)(17,89,62,39,42,67)(18,90,63,40,43,68)(19,91,64,33,44,69)(20,92,57,34,45,70)(21,93,58,35,46,71)(22,94,59,36,47,72)(23,95,60,37,48,65)(24,96,61,38,41,66), (1,18,16,40)(2,19,9,33)(3,20,10,34)(4,21,11,35)(5,22,12,36)(6,23,13,37)(7,24,14,38)(8,17,15,39)(25,59,84,72)(26,60,85,65)(27,61,86,66)(28,62,87,67)(29,63,88,68)(30,64,81,69)(31,57,82,70)(32,58,83,71)(41,75,96,56)(42,76,89,49)(43,77,90,50)(44,78,91,51)(45,79,92,52)(46,80,93,53)(47,73,94,54)(48,74,95,55), (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), (1,5)(2,4)(6,8)(9,11)(12,16)(13,15)(17,23)(18,22)(19,21)(25,50)(26,49)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,35)(36,40)(37,39)(41,61)(42,60)(43,59)(44,58)(45,57)(46,64)(47,63)(48,62)(65,89)(66,96)(67,95)(68,94)(69,93)(70,92)(71,91)(72,90)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,81) );
G=PermutationGroup([[(1,88,50,16,29,77),(2,81,51,9,30,78),(3,82,52,10,31,79),(4,83,53,11,32,80),(5,84,54,12,25,73),(6,85,55,13,26,74),(7,86,56,14,27,75),(8,87,49,15,28,76),(17,89,62,39,42,67),(18,90,63,40,43,68),(19,91,64,33,44,69),(20,92,57,34,45,70),(21,93,58,35,46,71),(22,94,59,36,47,72),(23,95,60,37,48,65),(24,96,61,38,41,66)], [(1,18,16,40),(2,19,9,33),(3,20,10,34),(4,21,11,35),(5,22,12,36),(6,23,13,37),(7,24,14,38),(8,17,15,39),(25,59,84,72),(26,60,85,65),(27,61,86,66),(28,62,87,67),(29,63,88,68),(30,64,81,69),(31,57,82,70),(32,58,83,71),(41,75,96,56),(42,76,89,49),(43,77,90,50),(44,78,91,51),(45,79,92,52),(46,80,93,53),(47,73,94,54),(48,74,95,55)], [(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)], [(1,5),(2,4),(6,8),(9,11),(12,16),(13,15),(17,23),(18,22),(19,21),(25,50),(26,49),(27,56),(28,55),(29,54),(30,53),(31,52),(32,51),(33,35),(36,40),(37,39),(41,61),(42,60),(43,59),(44,58),(45,57),(46,64),(47,63),(48,62),(65,89),(66,96),(67,95),(68,94),(69,93),(70,92),(71,91),(72,90),(73,88),(74,87),(75,86),(76,85),(77,84),(78,83),(79,82),(80,81)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | D8 | C4○D4 | C4×S3 | C4○D8 | Q8⋊3S3 | S3×D4 | S3×D8 | D24⋊C2 |
| kernel | Dic3⋊5D8 | C6.D8 | C8×Dic3 | C3×C2.D8 | Dic3⋊5D4 | C2×D24 | D24 | C2.D8 | C2×Dic3 | C4⋊C4 | C2×C8 | Dic3 | C12 | C8 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 1 | 2 | 2 | 1 | 4 | 2 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of Dic3⋊5D8 ►in GL5(𝔽73)
| 72 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 72 | 0 |
| 46 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 41 | 25 | 0 | 0 |
| 0 | 35 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 |
| 0 | 25 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,1,0],[46,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,41,35,0,0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,1],[72,0,0,0,0,0,72,25,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;
Dic3⋊5D8 in GAP, Magma, Sage, TeX
{\rm Dic}_3\rtimes_5D_8 % in TeX
G:=Group("Dic3:5D8"); // GroupNames label
G:=SmallGroup(192,431);
// by ID
G=gap.SmallGroup(192,431);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,120,135,100,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^8=d^2=1,b^2=a^3,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations