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G = D8.D5order 160 = 25·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D8.D5, C52SD32, C20.4D4, C10.9D8, C8.5D10, Dic203C2, C40.3C22, C52C162C2, (C5×D8).1C2, C2.5(D4⋊D5), C4.2(C5⋊D4), SmallGroup(160,34)

Series: Derived Chief Lower central Upper central

C1C40 — D8.D5
C1C5C10C20C40Dic20 — D8.D5
C5C10C20C40 — D8.D5
C1C2C4C8D8

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 >

8C2
4C22
20C4
8C10
2D4
10Q8
4Dic5
4C2×C10
5C16
5Q16
2Dic10
2C5×D4
5SD32

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
ρ6220-202200220000-222-2-2-20000    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
ρ11220-2022002200002-2-22-2-20000    orthogonal lifted from D8
ρ1222020-1-5/2-1+5/2-2-2-1-5/2-1+5/2ζ5352ζ54553525450000-1-5/2-1+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+5/2-1-5/2ζ5455352545ζ53520000-1+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+5/25352545ζ5352ζ5450000-1-5/2-1+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+5/2-1-5/2545ζ5352ζ54553520000-1+5/2-1-5/21+5/21+5/21-5/21-5/2    complex lifted from C5⋊D4
ρ162-200022-22-2-20000ζ16716ζ16131611ζ165163ζ1615169002-2-22    complex lifted from SD32
ρ172-2000222-2-2-20000ζ165163ζ16716ζ1615169ζ1613161100-222-2    complex lifted from SD32
ρ182-2000222-2-2-20000ζ16131611ζ1615169ζ16716ζ16516300-222-2    complex lifted from SD32
ρ192-200022-22-2-20000ζ1615169ζ165163ζ16131611ζ16716002-2-22    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+5-22221+51-50000000000ζ87ζ5487ζ585ζ5485ζ5ζ83ζ5483ζ58ζ548ζ5ζ83ζ5383ζ528ζ538ζ5283ζ5383ζ528ζ538ζ52    symplectic faithful, Schur index 2
ρ234-4000-1-5-1+522-221+51-50000000000ζ83ζ5483ζ58ζ548ζ5ζ87ζ5487ζ585ζ5485ζ583ζ5383ζ528ζ538ζ52ζ83ζ5383ζ528ζ538ζ52    symplectic faithful, Schur index 2
ρ244-4000-1+5-1-5-22221-51+5000000000083ζ5383ζ528ζ538ζ52ζ83ζ5383ζ528ζ538ζ52ζ83ζ5483ζ58ζ548ζ5ζ87ζ5487ζ585ζ5485ζ5    symplectic faithful, Schur index 2
ρ254-4000-1+5-1-522-221-51+50000000000ζ83ζ5383ζ528ζ538ζ5283ζ5383ζ528ζ538ζ52ζ87ζ5487ζ585ζ5485ζ5ζ83ζ5483ζ58ζ548ζ5    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)(18 24)(19 23)(20 22)(25 29)(26 28)(30 32)(34 40)(35 39)(36 38)(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 44 59)(2 54 72 45 60)(3 55 65 46 61)(4 56 66 47 62)(5 49 67 48 63)(6 50 68 41 64)(7 51 69 42 57)(8 52 70 43 58)(9 27 33 21 75)(10 28 34 22 76)(11 29 35 23 77)(12 30 36 24 78)(13 31 37 17 79)(14 32 38 18 80)(15 25 39 19 73)(16 26 40 20 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 41 21 45)(18 48 22 44)(19 47 23 43)(20 46 24 42)(33 60 37 64)(34 59 38 63)(35 58 39 62)(36 57 40 61)(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)(18,24)(19,23)(20,22)(25,29)(26,28)(30,32)(34,40)(35,39)(36,38)(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), (1,53,71,44,59)(2,54,72,45,60)(3,55,65,46,61)(4,56,66,47,62)(5,49,67,48,63)(6,50,68,41,64)(7,51,69,42,57)(8,52,70,43,58)(9,27,33,21,75)(10,28,34,22,76)(11,29,35,23,77)(12,30,36,24,78)(13,31,37,17,79)(14,32,38,18,80)(15,25,39,19,73)(16,26,40,20,74), (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)(17,41,21,45)(18,48,22,44)(19,47,23,43)(20,46,24,42)(33,60,37,64)(34,59,38,63)(35,58,39,62)(36,57,40,61)(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)(18,24)(19,23)(20,22)(25,29)(26,28)(30,32)(34,40)(35,39)(36,38)(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), (1,53,71,44,59)(2,54,72,45,60)(3,55,65,46,61)(4,56,66,47,62)(5,49,67,48,63)(6,50,68,41,64)(7,51,69,42,57)(8,52,70,43,58)(9,27,33,21,75)(10,28,34,22,76)(11,29,35,23,77)(12,30,36,24,78)(13,31,37,17,79)(14,32,38,18,80)(15,25,39,19,73)(16,26,40,20,74), (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)(17,41,21,45)(18,48,22,44)(19,47,23,43)(20,46,24,42)(33,60,37,64)(34,59,38,63)(35,58,39,62)(36,57,40,61)(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),(18,24),(19,23),(20,22),(25,29),(26,28),(30,32),(34,40),(35,39),(36,38),(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)], [(1,53,71,44,59),(2,54,72,45,60),(3,55,65,46,61),(4,56,66,47,62),(5,49,67,48,63),(6,50,68,41,64),(7,51,69,42,57),(8,52,70,43,58),(9,27,33,21,75),(10,28,34,22,76),(11,29,35,23,77),(12,30,36,24,78),(13,31,37,17,79),(14,32,38,18,80),(15,25,39,19,73),(16,26,40,20,74)], [(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),(17,41,21,45),(18,48,22,44),(19,47,23,43),(20,46,24,42),(33,60,37,64),(34,59,38,63),(35,58,39,62),(36,57,40,61),(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  D163D5  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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