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## G = SD16⋊D5order 160 = 25·5

### 2nd semidirect product of SD16 and D5 acting via D5/C5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — SD16⋊D5
 Chief series C1 — C5 — C10 — C20 — C4×D5 — Q8×D5 — SD16⋊D5
 Lower central C5 — C10 — C20 — SD16⋊D5
 Upper central C1 — C2 — C4 — SD16

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, C4, C4, C22, C5, C8, C8, C2×C4, D4, D4, Q8, Q8, D5, C10, C10, M4(2), SD16, SD16, Q16, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, C2×C10, C8.C22, C52C8, C40, Dic10, Dic10, C4×D5, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, C8⋊D5, Dic20, D4.D5, C5⋊Q16, C5×SD16, D42D5, Q8×D5, SD16⋊D5
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, 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

Smallest permutation representation of SD16⋊D5
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 36)(10 39)(11 34)(12 37)(13 40)(14 35)(15 38)(16 33)(17 30)(18 25)(19 28)(20 31)(21 26)(22 29)(23 32)(24 27)(41 67)(42 70)(43 65)(44 68)(45 71)(46 66)(47 69)(48 72)(49 80)(50 75)(51 78)(52 73)(53 76)(54 79)(55 74)(56 77)
(1 75 67 30 11)(2 76 68 31 12)(3 77 69 32 13)(4 78 70 25 14)(5 79 71 26 15)(6 80 72 27 16)(7 73 65 28 9)(8 74 66 29 10)(17 34 64 50 41)(18 35 57 51 42)(19 36 58 52 43)(20 37 59 53 44)(21 38 60 54 45)(22 39 61 55 46)(23 40 62 56 47)(24 33 63 49 48)
(1 11)(2 16)(3 13)(4 10)(5 15)(6 12)(7 9)(8 14)(17 54)(18 51)(19 56)(20 53)(21 50)(22 55)(23 52)(24 49)(25 74)(26 79)(27 76)(28 73)(29 78)(30 75)(31 80)(32 77)(33 63)(34 60)(35 57)(36 62)(37 59)(38 64)(39 61)(40 58)(41 45)(43 47)(66 70)(68 72)

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)(9,36)(10,39)(11,34)(12,37)(13,40)(14,35)(15,38)(16,33)(17,30)(18,25)(19,28)(20,31)(21,26)(22,29)(23,32)(24,27)(41,67)(42,70)(43,65)(44,68)(45,71)(46,66)(47,69)(48,72)(49,80)(50,75)(51,78)(52,73)(53,76)(54,79)(55,74)(56,77), (1,75,67,30,11)(2,76,68,31,12)(3,77,69,32,13)(4,78,70,25,14)(5,79,71,26,15)(6,80,72,27,16)(7,73,65,28,9)(8,74,66,29,10)(17,34,64,50,41)(18,35,57,51,42)(19,36,58,52,43)(20,37,59,53,44)(21,38,60,54,45)(22,39,61,55,46)(23,40,62,56,47)(24,33,63,49,48), (1,11)(2,16)(3,13)(4,10)(5,15)(6,12)(7,9)(8,14)(17,54)(18,51)(19,56)(20,53)(21,50)(22,55)(23,52)(24,49)(25,74)(26,79)(27,76)(28,73)(29,78)(30,75)(31,80)(32,77)(33,63)(34,60)(35,57)(36,62)(37,59)(38,64)(39,61)(40,58)(41,45)(43,47)(66,70)(68,72)>;

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)(9,36)(10,39)(11,34)(12,37)(13,40)(14,35)(15,38)(16,33)(17,30)(18,25)(19,28)(20,31)(21,26)(22,29)(23,32)(24,27)(41,67)(42,70)(43,65)(44,68)(45,71)(46,66)(47,69)(48,72)(49,80)(50,75)(51,78)(52,73)(53,76)(54,79)(55,74)(56,77), (1,75,67,30,11)(2,76,68,31,12)(3,77,69,32,13)(4,78,70,25,14)(5,79,71,26,15)(6,80,72,27,16)(7,73,65,28,9)(8,74,66,29,10)(17,34,64,50,41)(18,35,57,51,42)(19,36,58,52,43)(20,37,59,53,44)(21,38,60,54,45)(22,39,61,55,46)(23,40,62,56,47)(24,33,63,49,48), (1,11)(2,16)(3,13)(4,10)(5,15)(6,12)(7,9)(8,14)(17,54)(18,51)(19,56)(20,53)(21,50)(22,55)(23,52)(24,49)(25,74)(26,79)(27,76)(28,73)(29,78)(30,75)(31,80)(32,77)(33,63)(34,60)(35,57)(36,62)(37,59)(38,64)(39,61)(40,58)(41,45)(43,47)(66,70)(68,72) );

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),(9,36),(10,39),(11,34),(12,37),(13,40),(14,35),(15,38),(16,33),(17,30),(18,25),(19,28),(20,31),(21,26),(22,29),(23,32),(24,27),(41,67),(42,70),(43,65),(44,68),(45,71),(46,66),(47,69),(48,72),(49,80),(50,75),(51,78),(52,73),(53,76),(54,79),(55,74),(56,77)], [(1,75,67,30,11),(2,76,68,31,12),(3,77,69,32,13),(4,78,70,25,14),(5,79,71,26,15),(6,80,72,27,16),(7,73,65,28,9),(8,74,66,29,10),(17,34,64,50,41),(18,35,57,51,42),(19,36,58,52,43),(20,37,59,53,44),(21,38,60,54,45),(22,39,61,55,46),(23,40,62,56,47),(24,33,63,49,48)], [(1,11),(2,16),(3,13),(4,10),(5,15),(6,12),(7,9),(8,14),(17,54),(18,51),(19,56),(20,53),(21,50),(22,55),(23,52),(24,49),(25,74),(26,79),(27,76),(28,73),(29,78),(30,75),(31,80),(32,77),(33,63),(34,60),(35,57),(36,62),(37,59),(38,64),(39,61),(40,58),(41,45),(43,47),(66,70),(68,72)]])

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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