metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3⋊5SD16, (C3×Q8)⋊4D4, (Q8×Dic3)⋊4C2, (C2×SD16)⋊7S3, C3⋊5(C4⋊SD16), (C2×D4).68D6, Q8⋊2(C3⋊D4), (C2×C8).144D6, Dic3⋊C8⋊33C2, (C6×SD16)⋊18C2, C12⋊3D4.5C2, C12.171(C2×D4), C2.D24⋊33C2, (C2×Q8).137D6, C2.27(S3×SD16), C6.44(C2×SD16), C12.97(C4○D4), D4⋊Dic3⋊32C2, C2.26(Q8⋊3D6), C6.75(C8⋊C22), (C2×Dic3).67D4, (C6×D4).89C22, C22.261(S3×D4), (C6×Q8).70C22, C4.10(D4⋊2S3), C6.113(C4⋊D4), (C2×C12).440C23, (C2×C24).291C22, (C2×D12).117C22, C4⋊Dic3.170C22, (C4×Dic3).47C22, C2.25(C23.14D6), C4.39(C2×C3⋊D4), (C2×C6).352(C2×D4), (C2×Q8⋊2S3)⋊16C2, (C2×C3⋊C8).152C22, (C2×C4).529(C22×S3), SmallGroup(192,722)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3⋊5SD16
G = < a,b,c,d | a6=c8=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd=a3b, dcd=c3 >
Subgroups: 424 in 128 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, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C4⋊C4, C2×C8, C2×C8, SD16, C2×D4, C2×D4, C2×Q8, C3⋊C8, C24, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C3×Q8, C22×S3, C22×C6, D4⋊C4, C4⋊C8, C4×Q8, C4⋊1D4, C2×SD16, C2×SD16, C2×C3⋊C8, C4×Dic3, C4×Dic3, C4⋊Dic3, C4⋊Dic3, Q8⋊2S3, C2×C24, C3×SD16, C2×D12, C2×C3⋊D4, C6×D4, C6×Q8, C4⋊SD16, Dic3⋊C8, C2.D24, D4⋊Dic3, C12⋊3D4, C2×Q8⋊2S3, Q8×Dic3, C6×SD16, Dic3⋊5SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, C2×SD16, C8⋊C22, S3×D4, D4⋊2S3, C2×C3⋊D4, C4⋊SD16, S3×SD16, Q8⋊3D6, C23.14D6, Dic3⋊5SD16
►On 96 points
(1 73 30 14 49 83)(2 74 31 15 50 84)(3 75 32 16 51 85)(4 76 25 9 52 86)(5 77 26 10 53 87)(6 78 27 11 54 88)(7 79 28 12 55 81)(8 80 29 13 56 82)(17 89 63 35 43 67)(18 90 64 36 44 68)(19 91 57 37 45 69)(20 92 58 38 46 70)(21 93 59 39 47 71)(22 94 60 40 48 72)(23 95 61 33 41 65)(24 96 62 34 42 66)
(1 18 14 36)(2 37 15 19)(3 20 16 38)(4 39 9 21)(5 22 10 40)(6 33 11 23)(7 24 12 34)(8 35 13 17)(25 93 86 47)(26 48 87 94)(27 95 88 41)(28 42 81 96)(29 89 82 43)(30 44 83 90)(31 91 84 45)(32 46 85 92)(49 64 73 68)(50 69 74 57)(51 58 75 70)(52 71 76 59)(53 60 77 72)(54 65 78 61)(55 62 79 66)(56 67 80 63)
(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 22)(2 17)(3 20)(4 23)(5 18)(6 21)(7 24)(8 19)(9 33)(10 36)(11 39)(12 34)(13 37)(14 40)(15 35)(16 38)(25 61)(26 64)(27 59)(28 62)(29 57)(30 60)(31 63)(32 58)(41 52)(42 55)(43 50)(44 53)(45 56)(46 51)(47 54)(48 49)(65 86)(66 81)(67 84)(68 87)(69 82)(70 85)(71 88)(72 83)(73 94)(74 89)(75 92)(76 95)(77 90)(78 93)(79 96)(80 91)
G:=sub<Sym(96)| (1,73,30,14,49,83)(2,74,31,15,50,84)(3,75,32,16,51,85)(4,76,25,9,52,86)(5,77,26,10,53,87)(6,78,27,11,54,88)(7,79,28,12,55,81)(8,80,29,13,56,82)(17,89,63,35,43,67)(18,90,64,36,44,68)(19,91,57,37,45,69)(20,92,58,38,46,70)(21,93,59,39,47,71)(22,94,60,40,48,72)(23,95,61,33,41,65)(24,96,62,34,42,66), (1,18,14,36)(2,37,15,19)(3,20,16,38)(4,39,9,21)(5,22,10,40)(6,33,11,23)(7,24,12,34)(8,35,13,17)(25,93,86,47)(26,48,87,94)(27,95,88,41)(28,42,81,96)(29,89,82,43)(30,44,83,90)(31,91,84,45)(32,46,85,92)(49,64,73,68)(50,69,74,57)(51,58,75,70)(52,71,76,59)(53,60,77,72)(54,65,78,61)(55,62,79,66)(56,67,80,63), (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), (1,22)(2,17)(3,20)(4,23)(5,18)(6,21)(7,24)(8,19)(9,33)(10,36)(11,39)(12,34)(13,37)(14,40)(15,35)(16,38)(25,61)(26,64)(27,59)(28,62)(29,57)(30,60)(31,63)(32,58)(41,52)(42,55)(43,50)(44,53)(45,56)(46,51)(47,54)(48,49)(65,86)(66,81)(67,84)(68,87)(69,82)(70,85)(71,88)(72,83)(73,94)(74,89)(75,92)(76,95)(77,90)(78,93)(79,96)(80,91)>;
G:=Group( (1,73,30,14,49,83)(2,74,31,15,50,84)(3,75,32,16,51,85)(4,76,25,9,52,86)(5,77,26,10,53,87)(6,78,27,11,54,88)(7,79,28,12,55,81)(8,80,29,13,56,82)(17,89,63,35,43,67)(18,90,64,36,44,68)(19,91,57,37,45,69)(20,92,58,38,46,70)(21,93,59,39,47,71)(22,94,60,40,48,72)(23,95,61,33,41,65)(24,96,62,34,42,66), (1,18,14,36)(2,37,15,19)(3,20,16,38)(4,39,9,21)(5,22,10,40)(6,33,11,23)(7,24,12,34)(8,35,13,17)(25,93,86,47)(26,48,87,94)(27,95,88,41)(28,42,81,96)(29,89,82,43)(30,44,83,90)(31,91,84,45)(32,46,85,92)(49,64,73,68)(50,69,74,57)(51,58,75,70)(52,71,76,59)(53,60,77,72)(54,65,78,61)(55,62,79,66)(56,67,80,63), (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), (1,22)(2,17)(3,20)(4,23)(5,18)(6,21)(7,24)(8,19)(9,33)(10,36)(11,39)(12,34)(13,37)(14,40)(15,35)(16,38)(25,61)(26,64)(27,59)(28,62)(29,57)(30,60)(31,63)(32,58)(41,52)(42,55)(43,50)(44,53)(45,56)(46,51)(47,54)(48,49)(65,86)(66,81)(67,84)(68,87)(69,82)(70,85)(71,88)(72,83)(73,94)(74,89)(75,92)(76,95)(77,90)(78,93)(79,96)(80,91) );
G=PermutationGroup([[(1,73,30,14,49,83),(2,74,31,15,50,84),(3,75,32,16,51,85),(4,76,25,9,52,86),(5,77,26,10,53,87),(6,78,27,11,54,88),(7,79,28,12,55,81),(8,80,29,13,56,82),(17,89,63,35,43,67),(18,90,64,36,44,68),(19,91,57,37,45,69),(20,92,58,38,46,70),(21,93,59,39,47,71),(22,94,60,40,48,72),(23,95,61,33,41,65),(24,96,62,34,42,66)], [(1,18,14,36),(2,37,15,19),(3,20,16,38),(4,39,9,21),(5,22,10,40),(6,33,11,23),(7,24,12,34),(8,35,13,17),(25,93,86,47),(26,48,87,94),(27,95,88,41),(28,42,81,96),(29,89,82,43),(30,44,83,90),(31,91,84,45),(32,46,85,92),(49,64,73,68),(50,69,74,57),(51,58,75,70),(52,71,76,59),(53,60,77,72),(54,65,78,61),(55,62,79,66),(56,67,80,63)], [(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)], [(1,22),(2,17),(3,20),(4,23),(5,18),(6,21),(7,24),(8,19),(9,33),(10,36),(11,39),(12,34),(13,37),(14,40),(15,35),(16,38),(25,61),(26,64),(27,59),(28,62),(29,57),(30,60),(31,63),(32,58),(41,52),(42,55),(43,50),(44,53),(45,56),(46,51),(47,54),(48,49),(65,86),(66,81),(67,84),(68,87),(69,82),(70,85),(71,88),(72,83),(73,94),(74,89),(75,92),(76,95),(77,90),(78,93),(79,96),(80,91)]])
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 | C3⋊D4 | C8⋊C22 | D4⋊2S3 | S3×D4 | S3×SD16 | Q8⋊3D6 |
| kernel | Dic3⋊5SD16 | Dic3⋊C8 | C2.D24 | D4⋊Dic3 | C12⋊3D4 | C2×Q8⋊2S3 | Q8×Dic3 | C6×SD16 | C2×SD16 | C2×Dic3 | C3×Q8 | C2×C8 | C2×D4 | C2×Q8 | Dic3 | C12 | Q8 | 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 Dic3⋊5SD16 ►in GL4(𝔽73) generated by
| 1 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 43 | 0 | 0 |
| 30 | 13 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 30 | 13 | 0 | 0 |
| 60 | 43 | 0 | 0 |
| 0 | 0 | 61 | 55 |
| 0 | 0 | 4 | 0 |
| 43 | 60 | 0 | 0 |
| 13 | 30 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 48 | 72 |
G:=sub<GL(4,GF(73))| [1,1,0,0,72,0,0,0,0,0,1,0,0,0,0,1],[60,30,0,0,43,13,0,0,0,0,72,0,0,0,0,72],[30,60,0,0,13,43,0,0,0,0,61,4,0,0,55,0],[43,13,0,0,60,30,0,0,0,0,1,48,0,0,0,72] >;
Dic3⋊5SD16 in GAP, Magma, Sage, TeX
{\rm Dic}_3\rtimes_5{\rm SD}_{16} % in TeX
G:=Group("Dic3:5SD16"); // GroupNames label
G:=SmallGroup(192,722);
// by ID
G=gap.SmallGroup(192,722);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,135,184,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^8=d^2=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=a^3*b,d*c*d=c^3>;
// generators/relations