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G = D8.D5  order 160 = 25·5

The non-split extension by D8 of D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D8.D5, C5⋊2SD32, C20.4D4, C10.9D8, C8.5D10, Dic20⋊3C2, C40.3C22, C5⋊2C16⋊2C2, (C5×D8).1C2, C2.5(D4⋊D5), C4.2(C5⋊D4), SmallGroup(160,34)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C40 — D8.D5
Chief series
C1 — C5 — C10 — C20 — C40 — Dic20 — D8.D5
Lower central
C5 — C10 — C20 — C40 — D8.D5
Upper central
C1 — C2 — C4 — C8 — D8

Generators and relations for D8.D5
 G = < a,b,c,d | a8=b2=c5=1, d2=a4, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=a5b, dcd-1=c-1 >

C1
C2
8C2
C5
C4
4C22
20C4
C10
8C10
C8
2D4
10Q8
C20
4Dic5
4C2×C10
D8
5C16
5Q16
C40
2Dic10
2C5×D4
5SD32
C5⋊2C16
C5×D8
Dic20
D8.D5

Character table of D8.D5

 class 12A2B4A4B5A5B8A8B10A10B10C10D10E10F16A16B16C16D20A20B40A40B40C40D
 size 118240222222888810101010444444
ρ11111111111111111111111111    trivial
ρ21111-11111111111-1-1-1-1111111    linear of order 2
ρ311-11-1111111-1-1-1-11111111111    linear of order 2
ρ411-111111111-1-1-1-1-1-1-1-1111111    linear of order 2
ρ52202022-2-2220000000022-2-2-2-2    orthogonal lifted from D4
ρ6220-202200220000-√2√2√2-√2-2-20000    orthogonal lifted from D8
ρ722220-1+√5/2-1-√5/222-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/20000-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2    orthogonal lifted from D5
ρ822-220-1-√5/2-1+√5/222-1-√5/2-1+√5/21+√5/21-√5/21+√5/21-√5/20000-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2    orthogonal lifted from D10
ρ922220-1-√5/2-1+√5/222-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/20000-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2    orthogonal lifted from D5
ρ1022-220-1+√5/2-1-√5/222-1+√5/2-1-√5/21-√5/21+√5/21-√5/21+√5/20000-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2    orthogonal lifted from D10
ρ11220-202200220000√2-√2-√2√2-2-20000    orthogonal lifted from D8
ρ1222020-1-√5/2-1+√5/2-2-2-1-√5/2-1+√5/2ζ53-ζ52ζ54-ζ5-ζ53+ζ52-ζ54+ζ50000-1-√5/2-1+√5/21-√5/21-√5/21+√5/21+√5/2    complex lifted from C5⋊D4
ρ1322020-1+√5/2-1-√5/2-2-2-1+√5/2-1-√5/2ζ54-ζ5-ζ53+ζ52-ζ54+ζ5ζ53-ζ520000-1+√5/2-1-√5/21+√5/21+√5/21-√5/21-√5/2    complex lifted from C5⋊D4
ρ1422020-1-√5/2-1+√5/2-2-2-1-√5/2-1+√5/2-ζ53+ζ52-ζ54+ζ5ζ53-ζ52ζ54-ζ50000-1-√5/2-1+√5/21-√5/21-√5/21+√5/21+√5/2    complex lifted from C5⋊D4
ρ1522020-1+√5/2-1-√5/2-2-2-1+√5/2-1-√5/2-ζ54+ζ5ζ53-ζ52ζ54-ζ5-ζ53+ζ520000-1+√5/2-1-√5/21+√5/21+√5/21-√5/21-√5/2    complex lifted from C5⋊D4
ρ162-200022-√2√2-2-20000ζ167+ζ16ζ1613+ζ1611ζ165+ζ163ζ1615+ζ16900√2-√2-√2√2    complex lifted from SD32
ρ172-200022√2-√2-2-20000ζ165+ζ163ζ167+ζ16ζ1615+ζ169ζ1613+ζ161100-√2√2√2-√2    complex lifted from SD32
ρ182-200022√2-√2-2-20000ζ1613+ζ1611ζ1615+ζ169ζ167+ζ16ζ165+ζ16300-√2√2√2-√2    complex lifted from SD32
ρ192-200022-√2√2-2-20000ζ1615+ζ169ζ165+ζ163ζ1613+ζ1611ζ167+ζ1600√2-√2-√2√2    complex lifted from SD32
ρ20440-40-1+√5-1-√500-1+√5-1-√5000000001-√51+√50000    orthogonal lifted from D4⋊D5, Schur index 2
ρ21440-40-1-√5-1+√500-1-√5-1+√5000000001+√51-√50000    orthogonal lifted from D4⋊D5, Schur index 2
ρ224-4000-1-√5-1+√5-2√22√21+√51-√50000000000ζ87ζ54+ζ87ζ5-ζ85ζ54-ζ85ζ5ζ83ζ54+ζ83ζ5-ζ8ζ54-ζ8ζ5ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52-ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52    symplectic faithful, Schur index 2
ρ234-4000-1-√5-1+√52√2-2√21+√51-√50000000000ζ83ζ54+ζ83ζ5-ζ8ζ54-ζ8ζ5ζ87ζ54+ζ87ζ5-ζ85ζ54-ζ85ζ5-ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52    symplectic faithful, Schur index 2
ρ244-4000-1+√5-1-√5-2√22√21-√51+√50000000000-ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52ζ83ζ54+ζ83ζ5-ζ8ζ54-ζ8ζ5ζ87ζ54+ζ87ζ5-ζ85ζ54-ζ85ζ5    symplectic faithful, Schur index 2
ρ254-4000-1+√5-1-√52√2-2√21-√51+√50000000000ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52-ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52ζ87ζ54+ζ87ζ5-ζ85ζ54-ζ85ζ5ζ83ζ54+ζ83ζ5-ζ8ζ54-ζ8ζ5    symplectic faithful, Schur index 2

Smallest permutation representation of D8.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 8)(2 7)(3 6)(4 5)(10 16)(11 15)(12 14)(17 19)(20 24)(21 23)(25 29)(26 28)(30 32)(33 39)(34 38)(35 37)(41 46)(42 45)(43 44)(47 48)(49 56)(50 55)(51 54)(52 53)(57 60)(58 59)(61 64)(62 63)(65 68)(66 67)(69 72)(70 71)(73 77)(74 76)(78 80)

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

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

D8.D5 is a maximal subgroup of
 D16⋊D5  D16⋊3D5  D5×SD32  SD32⋊D5  D8.D10  C40.30C23  C40.31C23  C15⋊SD32  D24.D5  D8.D15
D8.D5 is a maximal quotient of
 C10.SD32  C10.Q32  C10.D16  C15⋊SD32  D24.D5  D8.D15

Matrix representation of D8.D5 ►in GL4(𝔽241) generated by

1123000
111100
002400
000240
,
1123000
23023000
002400
001951
,
1000
0100
00980
0016191
,
20010300
1034100
00142151
0021699
G:=sub<GL(4,GF(241))| [11,11,0,0,230,11,0,0,0,0,240,0,0,0,0,240],[11,230,0,0,230,230,0,0,0,0,240,195,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,98,161,0,0,0,91],[200,103,0,0,103,41,0,0,0,0,142,216,0,0,151,99] >;
 

D8.D5 in GAP, Magma, Sage, TeX 

D_8.D_5
 
% in TeX
 
G:=Group("D8.D5");
 
// GroupNames label
 
G:=SmallGroup(160,34);
 
// by ID
 
G=gap.SmallGroup(160,34);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,73,218,116,122,579,297,69,4613]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^5=1,d^2=a^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D8.D5 in TeX
Character table of D8.D5 in TeX

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