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G = SD16⋊D5order 160 = 25·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C8.2D10, SD162D5, D4.4D10, Q8.1D10, Dic206C2, D10.16D4, C40.9C22, C20.6C23, Dic5.18D4, Dic10.2C22, (Q8×D5)⋊2C2, C8⋊D52C2, D4.D54C2, C5⋊Q161C2, C2.20(D4×D5), (C5×SD16)⋊2C2, C10.32(C2×D4), C52(C8.C22), C4.6(C22×D5), D42D5.1C2, C52C8.1C22, (C5×D4).4C22, (C4×D5).3C22, (C5×Q8).1C22, SmallGroup(160,136)

Series: Derived Chief Lower central Upper central

C1C20 — SD16⋊D5
C1C5C10C20C4×D5Q8×D5 — SD16⋊D5
C5C10C20 — SD16⋊D5
C1C2C4SD16

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, C52C8, 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, D42D5, 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 12A2B2C4A4B4C4D4E5A5B8A8B10A10B10C10D20A20B20C20D40A40B40C40D
 size 114102410202022420228844884444
ρ11111111111111111111111111    trivial
ρ211-1-111-1-1111-1111-1-11111-1-1-1-1    linear of order 2
ρ3111-111-1-1-1111-1111111111111    linear of order 2
ρ411-111111-111-1-111-1-11111-1-1-1-1    linear of order 2
ρ511-111-11-1-1111111-1-111-1-11111    linear of order 2
ρ6111-11-1-11-111-11111111-1-1-1-1-1-1    linear of order 2
ρ711-1-11-1-111111-111-1-111-1-11111    linear of order 2
ρ811111-11-1111-1-1111111-1-1-1-1-1-1    linear of order 2
ρ9220-2-2020022002200-2-2000000    orthogonal lifted from D4
ρ102202-20-20022002200-2-2000000    orthogonal lifted from D4
ρ1122-2022000-1+5/2-1-5/2-20-1+5/2-1-5/21-5/21+5/2-1-5/2-1+5/2-1-5/2-1+5/21+5/21+5/21-5/21-5/2    orthogonal lifted from D10
ρ1222-202-2000-1+5/2-1-5/220-1+5/2-1-5/21-5/21+5/2-1-5/2-1+5/21+5/21-5/2-1-5/2-1-5/2-1+5/2-1+5/2    orthogonal lifted from D10
ρ1322202-2000-1-5/2-1+5/2-20-1-5/2-1+5/2-1-5/2-1+5/2-1+5/2-1-5/21-5/21+5/21-5/21-5/21+5/21+5/2    orthogonal lifted from D10
ρ14222022000-1-5/2-1+5/220-1-5/2-1+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
ρ15222022000-1+5/2-1-5/220-1+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
ρ1622202-2000-1+5/2-1-5/2-20-1+5/2-1-5/2-1+5/2-1-5/2-1-5/2-1+5/21+5/21-5/21+5/21+5/21-5/21-5/2    orthogonal lifted from D10
ρ1722-202-2000-1-5/2-1+5/220-1-5/2-1+5/21+5/21-5/2-1+5/2-1-5/21-5/21+5/2-1+5/2-1+5/2-1-5/2-1-5/2    orthogonal lifted from D10
ρ1822-2022000-1-5/2-1+5/2-20-1-5/2-1+5/21+5/21-5/2-1+5/2-1-5/2-1+5/2-1-5/21-5/21-5/21+5/21+5/2    orthogonal lifted from D10
ρ194400-40000-1-5-1+500-1-5-1+5001-51+5000000    orthogonal lifted from D4×D5
ρ204400-40000-1+5-1-500-1+5-1-5001+51-5000000    orthogonal lifted from D4×D5
ρ214-400000004400-4-40000000000    symplectic lifted from C8.C22, Schur index 2
ρ224-40000000-1+5-1-5001-51+500000083ζ5383ζ528ζ538ζ52ζ83ζ5383ζ528ζ538ζ5287ζ5487ζ585ζ5485ζ583ζ5483ζ58ζ548ζ5    symplectic faithful, Schur index 2
ρ234-40000000-1+5-1-5001-51+5000000ζ83ζ5383ζ528ζ538ζ5283ζ5383ζ528ζ538ζ5283ζ5483ζ58ζ548ζ587ζ5487ζ585ζ5485ζ5    symplectic faithful, Schur index 2
ρ244-40000000-1-5-1+5001+51-500000087ζ5487ζ585ζ5485ζ583ζ5483ζ58ζ548ζ5ζ83ζ5383ζ528ζ538ζ5283ζ5383ζ528ζ538ζ52    symplectic faithful, Schur index 2
ρ254-40000000-1-5-1+5001+51-500000083ζ5483ζ58ζ548ζ587ζ5487ζ585ζ5485ζ583ζ5383ζ528ζ538ζ52ζ83ζ5383ζ528ζ538ζ52    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 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  D86D10  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.D55C4  Dic5.14D8  C20⋊Q8⋊C2  Dic10.D4  D4⋊(C4×D5)  D10.12D8  C52C8⋊D4  D4.D20  C5⋊Q165C4  Dic5⋊Q16  Q8.Dic10  C408C4.C2  (Q8×D5)⋊C4  D104Q16  (C2×C8).D10  C52C8.D4  Dic2015C4  Dic10⋊Q8  C403Q8  Dic10.Q8  C8⋊(C4×D5)  D10.12SD16  C20.(C4○D4)  C8.2D20  Dic53SD16  SD16⋊Dic5  (C5×Q8).D4  C40.31D4  D108SD16  Dic10.16D4  C408D4  Dic60⋊C2  D30.4D4  C60.8C23  D30.9D4  D12.27D10  D30.44D4  SD16⋊D15

Matrix representation of SD16⋊D5 in GL4(𝔽41) generated by

603535
110030
23151811
24262417
,
38221717
3127038
373519
2193912
,
40100
53500
40001
614034
,
40000
5100
40001
40010
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

Export

Character table of SD16⋊D5 in TeX

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