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G = D83D5order 160 = 25·5

The semidirect product of D8 and D5 acting through Inn(D8)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D83D5, C8.8D10, D4.1D10, D10.5D4, Dic204C2, C20.3C23, C40.6C22, Dic5.24D4, Dic10.1C22, (C8×D5)⋊2C2, (C5×D8)⋊3C2, C52(C4○D8), D4.D52C2, C2.17(D4×D5), D42D52C2, C10.29(C2×D4), C4.3(C22×D5), C52C8.5C22, (C5×D4).1C22, (C4×D5).16C22, SmallGroup(160,133)

Series: Derived Chief Lower central Upper central

C1C20 — D83D5
C1C5C10C20C4×D5D42D5 — D83D5
C5C10C20 — D83D5
C1C2C4D8

Generators and relations for D83D5
 G = < a,b,c,d | a8=b2=c5=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 200 in 62 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2 [×3], C4, C4 [×3], C22 [×3], C5, C8, C8, C2×C4 [×3], D4 [×2], D4 [×2], Q8 [×2], D5, C10, C10 [×2], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], Dic5, Dic5 [×2], C20, D10, C2×C10 [×2], C4○D8, C52C8, C40, Dic10 [×2], C4×D5, C2×Dic5 [×2], C5⋊D4 [×2], C5×D4 [×2], C8×D5, Dic20, D4.D5 [×2], C5×D8, D42D5 [×2], D83D5
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, D10 [×3], C4○D8, C22×D5, D4×D5, D83D5

Character table of D83D5

 class 12A2B2C2D4A4B4C4D4E5A5B8A8B8C8D10A10B10C10D10E10F20A20B40A40B40C40D
 size 114410255202022221010228888444444
ρ11111111111111111111111111111    trivial
ρ211-1-11111-1-111111111-1-1-1-1111111    linear of order 2
ρ311-11-11-1-11-111-1-11111-111-111-1-1-1-1    linear of order 2
ρ4111-1-11-1-1-1111-1-111111-1-1111-1-1-1-1    linear of order 2
ρ5111-111111-111-1-1-1-1111-1-1111-1-1-1-1    linear of order 2
ρ611-111111-1111-1-1-1-111-111-111-1-1-1-1    linear of order 2
ρ711-1-1-11-1-1111111-1-111-1-1-1-1111111    linear of order 2
ρ81111-11-1-1-1-11111-1-1111111111111    linear of order 2
ρ922002-2-2-200220000220000-2-20000    orthogonal lifted from D4
ρ102200-2-22200220000220000-2-20000    orthogonal lifted from D4
ρ1122-22020000-1+5/2-1-5/2-2-200-1-5/2-1+5/21-5/2-1+5/2-1-5/21+5/2-1-5/2-1+5/21-5/21+5/21-5/21+5/2    orthogonal lifted from D10
ρ1222-22020000-1-5/2-1+5/2-2-200-1+5/2-1-5/21+5/2-1-5/2-1+5/21-5/2-1+5/2-1-5/21+5/21-5/21+5/21-5/2    orthogonal lifted from D10
ρ1322-2-2020000-1+5/2-1-5/22200-1-5/2-1+5/21-5/21-5/21+5/21+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1+5/2-1-5/2    orthogonal lifted from D10
ρ142222020000-1-5/2-1+5/22200-1+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
ρ152222020000-1+5/2-1-5/22200-1-5/2-1+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
ρ1622-2-2020000-1-5/2-1+5/22200-1+5/2-1-5/21+5/21+5/21-5/21-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1-5/2-1+5/2    orthogonal lifted from D10
ρ17222-2020000-1+5/2-1-5/2-2-200-1-5/2-1+5/2-1+5/21-5/21+5/2-1-5/2-1-5/2-1+5/21-5/21+5/21-5/21+5/2    orthogonal lifted from D10
ρ18222-2020000-1-5/2-1+5/2-2-200-1+5/2-1-5/2-1-5/21+5/21-5/2-1+5/2-1+5/2-1-5/21+5/21-5/21+5/21-5/2    orthogonal lifted from D10
ρ192-20000-2i2i0022-22-2--2-2-20000002-2-22    complex lifted from C4○D8
ρ202-200002i-2i00222-2-2--2-2-2000000-222-2    complex lifted from C4○D8
ρ212-200002i-2i0022-22--2-2-2-20000002-2-22    complex lifted from C4○D8
ρ222-20000-2i2i00222-2--2-2-2-2000000-222-2    complex lifted from C4○D8
ρ2344000-40000-1+5-1-50000-1-5-1+500001+51-50000    orthogonal lifted from D4×D5
ρ2444000-40000-1-5-1+50000-1+5-1-500001-51+50000    orthogonal lifted from D4×D5
ρ254-400000000-1+5-1-5-2222001+51-5000000ζ87ζ5487ζ585ζ5485ζ5ζ83ζ5383ζ528ζ538ζ52ζ83ζ5483ζ58ζ548ζ583ζ5383ζ528ζ538ζ52    symplectic faithful, Schur index 2
ρ264-400000000-1-5-1+522-22001-51+5000000ζ83ζ5383ζ528ζ538ζ52ζ87ζ5487ζ585ζ5485ζ583ζ5383ζ528ζ538ζ52ζ83ζ5483ζ58ζ548ζ5    symplectic faithful, Schur index 2
ρ274-400000000-1-5-1+5-2222001-51+500000083ζ5383ζ528ζ538ζ52ζ83ζ5483ζ58ζ548ζ5ζ83ζ5383ζ528ζ538ζ52ζ87ζ5487ζ585ζ5485ζ5    symplectic faithful, Schur index 2
ρ284-400000000-1+5-1-522-22001+51-5000000ζ83ζ5483ζ58ζ548ζ583ζ5383ζ528ζ538ζ52ζ87ζ5487ζ585ζ5485ζ5ζ83ζ5383ζ528ζ538ζ52    symplectic faithful, Schur index 2

Smallest permutation representation of D83D5
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 59)(2 58)(3 57)(4 64)(5 63)(6 62)(7 61)(8 60)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 40)(16 39)(17 28)(18 27)(19 26)(20 25)(21 32)(22 31)(23 30)(24 29)(41 73)(42 80)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)(49 70)(50 69)(51 68)(52 67)(53 66)(54 65)(55 72)(56 71)
(1 66 77 29 10)(2 67 78 30 11)(3 68 79 31 12)(4 69 80 32 13)(5 70 73 25 14)(6 71 74 26 15)(7 72 75 27 16)(8 65 76 28 9)(17 38 60 54 46)(18 39 61 55 47)(19 40 62 56 48)(20 33 63 49 41)(21 34 64 50 42)(22 35 57 51 43)(23 36 58 52 44)(24 37 59 53 45)
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 49)(25 70)(26 71)(27 72)(28 65)(29 66)(30 67)(31 68)(32 69)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 57)(40 58)(41 45)(42 46)(43 47)(44 48)

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

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

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

D83D5 is a maximal subgroup of
D10.D8  D8.F5  D16⋊D5  D163D5  SD32⋊D5  SD323D5  D85F5  D8⋊F5  D813D10  D5×C4○D8  D20.47D4  SD16⋊D10  D86D10  D247D5  D245D5  D12.24D10  D30.11D4  D83D15
D83D5 is a maximal quotient of
Dic56SD16  D4.2Dic10  Dic10.D4  (C8×Dic5)⋊C2  D42D5⋊C4  D10⋊SD16  D4.D20  C405C4⋊C2  Dic55Q16  Dic10.2Q8  C8.6Dic10  C8.27(C4×D5)  D102Q16  C2.D87D5  D8×Dic5  (C2×D8).D5  C40.22D4  C406D4  Dic10⋊D4  D247D5  D245D5  D12.24D10  D30.11D4  D83D15

Matrix representation of D83D5 in GL4(𝔽41) generated by

27000
03800
00400
00040
,
03800
27000
0010
0001
,
1000
0100
0061
00400
,
1000
04000
0016
00040
G:=sub<GL(4,GF(41))| [27,0,0,0,0,38,0,0,0,0,40,0,0,0,0,40],[0,27,0,0,38,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,6,40,0,0,1,0],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,6,40] >;

D83D5 in GAP, Magma, Sage, TeX

D_8\rtimes_3D_5
% in TeX

G:=Group("D8:3D5");
// GroupNames label

G:=SmallGroup(160,133);
// by ID

G=gap.SmallGroup(160,133);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,362,116,297,159,69,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^5=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations

Export

Character table of D83D5 in TeX

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