metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8.2D10, SD16⋊2D5, D4.4D10, Q8.1D10, Dic20⋊6C2, D10.16D4, C40.9C22, C20.6C23, Dic5.18D4, Dic10.2C22, (Q8×D5)⋊2C2, C8⋊D5⋊2C2, D4.D5⋊4C2, C5⋊Q16⋊1C2, C2.20(D4×D5), (C5×SD16)⋊2C2, C10.32(C2×D4), C5⋊2(C8.C22), C4.6(C22×D5), D4⋊2D5.1C2, C5⋊2C8.1C22, (C5×D4).4C22, (C4×D5).3C22, (C5×Q8).1C22, SmallGroup(160,136)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD16⋊D5
G = < a,b,c,d | a8=b2=c5=d2=1, bab=a3, ac=ca, dad=a5, bc=cb, dbd=a4b, dcd=c-1 >
Subgroups: 192 in 60 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2 [×2], C4, C4 [×4], C22 [×2], C5, C8, C8, C2×C4 [×3], D4, D4, Q8, Q8 [×3], D5, C10, C10, M4(2), SD16, SD16, Q16 [×2], C2×Q8, C4○D4, Dic5, Dic5 [×2], C20, C20, D10, C2×C10, C8.C22, C5⋊2C8, C40, Dic10 [×2], Dic10, C4×D5, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, C8⋊D5, Dic20, D4.D5, C5⋊Q16, C5×SD16, D4⋊2D5, Q8×D5, SD16⋊D5
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, D10 [×3], C8.C22, C22×D5, D4×D5, SD16⋊D5
Character table of SD16⋊D5
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 10A | 10B | 10C | 10D | 20A | 20B | 20C | 20D | 40A | 40B | 40C | 40D | |
| size | 1 | 1 | 4 | 10 | 2 | 4 | 10 | 20 | 20 | 2 | 2 | 4 | 20 | 2 | 2 | 8 | 8 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ9 | 2 | 2 | 0 | -2 | -2 | 0 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 0 | 2 | -2 | 0 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -2 | 0 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | orthogonal lifted from D10 |
| ρ12 | 2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | 0 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | orthogonal lifted from D10 |
| ρ13 | 2 | 2 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -2 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | orthogonal lifted from D10 |
| ρ14 | 2 | 2 | 2 | 0 | 2 | 2 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ15 | 2 | 2 | 2 | 0 | 2 | 2 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ16 | 2 | 2 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -2 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | orthogonal lifted from D10 |
| ρ17 | 2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | 0 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | orthogonal lifted from D10 |
| ρ18 | 2 | 2 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -2 | 0 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | orthogonal lifted from D10 |
| ρ19 | 4 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4×D5 |
| ρ20 | 4 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4×D5 |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 4 | 0 | 0 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ83ζ53+ζ83ζ52-ζ8ζ53+ζ8ζ52 | ζ83ζ53-ζ83ζ52+ζ8ζ53-ζ8ζ52 | -ζ87ζ54+ζ87ζ5-ζ85ζ54+ζ85ζ5 | -ζ83ζ54+ζ83ζ5-ζ8ζ54+ζ8ζ5 | symplectic faithful, Schur index 2 |
| ρ23 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | ζ83ζ53-ζ83ζ52+ζ8ζ53-ζ8ζ52 | -ζ83ζ53+ζ83ζ52-ζ8ζ53+ζ8ζ52 | -ζ83ζ54+ζ83ζ5-ζ8ζ54+ζ8ζ5 | -ζ87ζ54+ζ87ζ5-ζ85ζ54+ζ85ζ5 | symplectic faithful, Schur index 2 |
| ρ24 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ87ζ54+ζ87ζ5-ζ85ζ54+ζ85ζ5 | -ζ83ζ54+ζ83ζ5-ζ8ζ54+ζ8ζ5 | ζ83ζ53-ζ83ζ52+ζ8ζ53-ζ8ζ52 | -ζ83ζ53+ζ83ζ52-ζ8ζ53+ζ8ζ52 | symplectic faithful, Schur index 2 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ83ζ54+ζ83ζ5-ζ8ζ54+ζ8ζ5 | -ζ87ζ54+ζ87ζ5-ζ85ζ54+ζ85ζ5 | -ζ83ζ53+ζ83ζ52-ζ8ζ53+ζ8ζ52 | ζ83ζ53-ζ83ζ52+ζ8ζ53-ζ8ζ52 | symplectic faithful, Schur index 2 |
►On 80 points
Generators in S80
(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)
(1 64)(2 59)(3 62)(4 57)(5 60)(6 63)(7 58)(8 61)(9 31)(10 26)(11 29)(12 32)(13 27)(14 30)(15 25)(16 28)(17 48)(18 43)(19 46)(20 41)(21 44)(22 47)(23 42)(24 45)(33 71)(34 66)(35 69)(36 72)(37 67)(38 70)(39 65)(40 68)(49 80)(50 75)(51 78)(52 73)(53 76)(54 79)(55 74)(56 77)
(1 75 9 48 66)(2 76 10 41 67)(3 77 11 42 68)(4 78 12 43 69)(5 79 13 44 70)(6 80 14 45 71)(7 73 15 46 72)(8 74 16 47 65)(17 34 64 50 31)(18 35 57 51 32)(19 36 58 52 25)(20 37 59 53 26)(21 38 60 54 27)(22 39 61 55 28)(23 40 62 56 29)(24 33 63 49 30)
(1 66)(2 71)(3 68)(4 65)(5 70)(6 67)(7 72)(8 69)(10 14)(12 16)(17 54)(18 51)(19 56)(20 53)(21 50)(22 55)(23 52)(24 49)(25 29)(27 31)(33 63)(34 60)(35 57)(36 62)(37 59)(38 64)(39 61)(40 58)(41 80)(42 77)(43 74)(44 79)(45 76)(46 73)(47 78)(48 75)
G:=sub<Sym(80)| (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), (1,64)(2,59)(3,62)(4,57)(5,60)(6,63)(7,58)(8,61)(9,31)(10,26)(11,29)(12,32)(13,27)(14,30)(15,25)(16,28)(17,48)(18,43)(19,46)(20,41)(21,44)(22,47)(23,42)(24,45)(33,71)(34,66)(35,69)(36,72)(37,67)(38,70)(39,65)(40,68)(49,80)(50,75)(51,78)(52,73)(53,76)(54,79)(55,74)(56,77), (1,75,9,48,66)(2,76,10,41,67)(3,77,11,42,68)(4,78,12,43,69)(5,79,13,44,70)(6,80,14,45,71)(7,73,15,46,72)(8,74,16,47,65)(17,34,64,50,31)(18,35,57,51,32)(19,36,58,52,25)(20,37,59,53,26)(21,38,60,54,27)(22,39,61,55,28)(23,40,62,56,29)(24,33,63,49,30), (1,66)(2,71)(3,68)(4,65)(5,70)(6,67)(7,72)(8,69)(10,14)(12,16)(17,54)(18,51)(19,56)(20,53)(21,50)(22,55)(23,52)(24,49)(25,29)(27,31)(33,63)(34,60)(35,57)(36,62)(37,59)(38,64)(39,61)(40,58)(41,80)(42,77)(43,74)(44,79)(45,76)(46,73)(47,78)(48,75)>;
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), (1,64)(2,59)(3,62)(4,57)(5,60)(6,63)(7,58)(8,61)(9,31)(10,26)(11,29)(12,32)(13,27)(14,30)(15,25)(16,28)(17,48)(18,43)(19,46)(20,41)(21,44)(22,47)(23,42)(24,45)(33,71)(34,66)(35,69)(36,72)(37,67)(38,70)(39,65)(40,68)(49,80)(50,75)(51,78)(52,73)(53,76)(54,79)(55,74)(56,77), (1,75,9,48,66)(2,76,10,41,67)(3,77,11,42,68)(4,78,12,43,69)(5,79,13,44,70)(6,80,14,45,71)(7,73,15,46,72)(8,74,16,47,65)(17,34,64,50,31)(18,35,57,51,32)(19,36,58,52,25)(20,37,59,53,26)(21,38,60,54,27)(22,39,61,55,28)(23,40,62,56,29)(24,33,63,49,30), (1,66)(2,71)(3,68)(4,65)(5,70)(6,67)(7,72)(8,69)(10,14)(12,16)(17,54)(18,51)(19,56)(20,53)(21,50)(22,55)(23,52)(24,49)(25,29)(27,31)(33,63)(34,60)(35,57)(36,62)(37,59)(38,64)(39,61)(40,58)(41,80)(42,77)(43,74)(44,79)(45,76)(46,73)(47,78)(48,75) );
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)], [(1,64),(2,59),(3,62),(4,57),(5,60),(6,63),(7,58),(8,61),(9,31),(10,26),(11,29),(12,32),(13,27),(14,30),(15,25),(16,28),(17,48),(18,43),(19,46),(20,41),(21,44),(22,47),(23,42),(24,45),(33,71),(34,66),(35,69),(36,72),(37,67),(38,70),(39,65),(40,68),(49,80),(50,75),(51,78),(52,73),(53,76),(54,79),(55,74),(56,77)], [(1,75,9,48,66),(2,76,10,41,67),(3,77,11,42,68),(4,78,12,43,69),(5,79,13,44,70),(6,80,14,45,71),(7,73,15,46,72),(8,74,16,47,65),(17,34,64,50,31),(18,35,57,51,32),(19,36,58,52,25),(20,37,59,53,26),(21,38,60,54,27),(22,39,61,55,28),(23,40,62,56,29),(24,33,63,49,30)], [(1,66),(2,71),(3,68),(4,65),(5,70),(6,67),(7,72),(8,69),(10,14),(12,16),(17,54),(18,51),(19,56),(20,53),(21,50),(22,55),(23,52),(24,49),(25,29),(27,31),(33,63),(34,60),(35,57),(36,62),(37,59),(38,64),(39,61),(40,58),(41,80),(42,77),(43,74),(44,79),(45,76),(46,73),(47,78),(48,75)])
SD16⋊D5 is a maximal subgroup of
D20.29D4 Q16⋊D10 D20.47D4 SD16⋊D10 D8⋊6D10 D5×C8.C22 D20.44D4 Dic60⋊C2 D30.4D4 C60.8C23 D30.9D4 D12.27D10 D30.44D4 SD16⋊D15 D10.1S4
SD16⋊D5 is a maximal quotient of
D4.D5⋊5C4 Dic5.14D8 C20⋊Q8⋊C2 Dic10.D4 D4⋊(C4×D5) D10.12D8 C5⋊2C8⋊D4 D4.D20 C5⋊Q16⋊5C4 Dic5⋊Q16 Q8.Dic10 C40⋊8C4.C2 (Q8×D5)⋊C4 D10⋊4Q16 (C2×C8).D10 C5⋊2C8.D4 Dic20⋊15C4 Dic10⋊Q8 C40⋊3Q8 Dic10.Q8 C8⋊(C4×D5) D10.12SD16 C20.(C4○D4) C8.2D20 Dic5⋊3SD16 SD16⋊Dic5 (C5×Q8).D4 C40.31D4 D10⋊8SD16 Dic10.16D4 C40⋊8D4 Dic60⋊C2 D30.4D4 C60.8C23 D30.9D4 D12.27D10 D30.44D4 SD16⋊D15
Matrix representation of SD16⋊D5 ►in GL4(𝔽41) generated by
| 6 | 0 | 35 | 35 |
| 11 | 0 | 0 | 30 |
| 23 | 15 | 18 | 11 |
| 24 | 26 | 24 | 17 |
| 38 | 22 | 17 | 17 |
| 31 | 27 | 0 | 38 |
| 37 | 3 | 5 | 19 |
| 2 | 19 | 39 | 12 |
| 40 | 1 | 0 | 0 |
| 5 | 35 | 0 | 0 |
| 40 | 0 | 0 | 1 |
| 6 | 1 | 40 | 34 |
| 40 | 0 | 0 | 0 |
| 5 | 1 | 0 | 0 |
| 40 | 0 | 0 | 1 |
| 40 | 0 | 1 | 0 |
G:=sub<GL(4,GF(41))| [6,11,23,24,0,0,15,26,35,0,18,24,35,30,11,17],[38,31,37,2,22,27,3,19,17,0,5,39,17,38,19,12],[40,5,40,6,1,35,0,1,0,0,0,40,0,0,1,34],[40,5,40,40,0,1,0,0,0,0,0,1,0,0,1,0] >;
SD16⋊D5 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\rtimes D_5 % in TeX
G:=Group("SD16:D5"); // GroupNames label
G:=SmallGroup(160,136);
// by ID
G=gap.SmallGroup(160,136);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,362,116,297,159,69,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^5=d^2=1,b*a*b=a^3,a*c=c*a,d*a*d=a^5,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations
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