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G = D4⋊D5order 80 = 24·5

The semidirect product of D4 and D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4⋊D5, C52D8, D202C2, C4.1D10, C10.7D4, C20.1C22, C52C81C2, (C5×D4)⋊1C2, C2.4(C5⋊D4), SmallGroup(80,15)

Series: Derived Chief Lower central Upper central

C1C20 — D4⋊D5
C1C5C10C20D20 — D4⋊D5
C5C10C20 — D4⋊D5
C1C2C4D4

Generators and relations for D4⋊D5
 G = < a,b,c,d | a4=b2=c5=d2=1, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=c-1 >

4C2
20C2
2C22
10C22
4D5
4C10
5C8
5D4
2D10
2C2×C10
5D8

Character table of D4⋊D5

 class 12A2B2C45A5B8A8B10A10B10C10D10E10F20A20B
 size 11420222101022444444
ρ111111111111111111    trivial
ρ211-1-11111111-1-1-1-111    linear of order 2
ρ3111-1111-1-111111111    linear of order 2
ρ411-11111-1-111-1-1-1-111    linear of order 2
ρ52200-22200220000-2-2    orthogonal lifted from D4
ρ622202-1+5/2-1-5/200-1+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
ρ722-202-1+5/2-1-5/200-1+5/2-1-5/21-5/21+5/21-5/21+5/2-1+5/2-1-5/2    orthogonal lifted from D10
ρ822202-1-5/2-1+5/200-1-5/2-1+5/2-1-5/2-1+5/2-1-5/2-1+5/2-1-5/2-1+5/2    orthogonal lifted from D5
ρ922-202-1-5/2-1+5/200-1-5/2-1+5/21+5/21-5/21+5/21-5/2-1-5/2-1+5/2    orthogonal lifted from D10
ρ102-200022-22-2-2000000    orthogonal lifted from D8
ρ112-2000222-2-2-2000000    orthogonal lifted from D8
ρ122200-2-1-5/2-1+5/200-1-5/2-1+5/2ζ53525455352ζ5451+5/21-5/2    complex lifted from C5⋊D4
ρ132200-2-1+5/2-1-5/200-1+5/2-1-5/25455352ζ545ζ53521-5/21+5/2    complex lifted from C5⋊D4
ρ142200-2-1+5/2-1-5/200-1+5/2-1-5/2ζ545ζ535254553521-5/21+5/2    complex lifted from C5⋊D4
ρ152200-2-1-5/2-1+5/200-1-5/2-1+5/25352ζ545ζ53525451+5/21-5/2    complex lifted from C5⋊D4
ρ164-4000-1+5-1-5001-51+5000000    orthogonal faithful, Schur index 2
ρ174-4000-1-5-1+5001+51-5000000    orthogonal faithful, Schur index 2

Smallest permutation representation of D4⋊D5
On 40 points
Generators in S40
(1 19 9 14)(2 20 10 15)(3 16 6 11)(4 17 7 12)(5 18 8 13)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)
(1 39)(2 40)(3 36)(4 37)(5 38)(6 31)(7 32)(8 33)(9 34)(10 35)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)
(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)
(1 5)(2 4)(7 10)(8 9)(11 16)(12 20)(13 19)(14 18)(15 17)(21 31)(22 35)(23 34)(24 33)(25 32)(26 36)(27 40)(28 39)(29 38)(30 37)

G:=sub<Sym(40)| (1,19,9,14)(2,20,10,15)(3,16,6,11)(4,17,7,12)(5,18,8,13)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (1,39)(2,40)(3,36)(4,37)(5,38)(6,31)(7,32)(8,33)(9,34)(10,35)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30), (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), (1,5)(2,4)(7,10)(8,9)(11,16)(12,20)(13,19)(14,18)(15,17)(21,31)(22,35)(23,34)(24,33)(25,32)(26,36)(27,40)(28,39)(29,38)(30,37)>;

G:=Group( (1,19,9,14)(2,20,10,15)(3,16,6,11)(4,17,7,12)(5,18,8,13)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (1,39)(2,40)(3,36)(4,37)(5,38)(6,31)(7,32)(8,33)(9,34)(10,35)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30), (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), (1,5)(2,4)(7,10)(8,9)(11,16)(12,20)(13,19)(14,18)(15,17)(21,31)(22,35)(23,34)(24,33)(25,32)(26,36)(27,40)(28,39)(29,38)(30,37) );

G=PermutationGroup([(1,19,9,14),(2,20,10,15),(3,16,6,11),(4,17,7,12),(5,18,8,13),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40)], [(1,39),(2,40),(3,36),(4,37),(5,38),(6,31),(7,32),(8,33),(9,34),(10,35),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30)], [(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)], [(1,5),(2,4),(7,10),(8,9),(11,16),(12,20),(13,19),(14,18),(15,17),(21,31),(22,35),(23,34),(24,33),(25,32),(26,36),(27,40),(28,39),(29,38),(30,37)])

D4⋊D5 is a maximal subgroup of
D5×D8  D8⋊D5  D40⋊C2  SD163D5  D4.D10  D4⋊D10  D4.8D10  C15⋊D8  C5⋊D24  D4⋊D15  D4⋊D25  C522D8  C5⋊D40  C527D8
D4⋊D5 is a maximal quotient of
C10.D8  D206C4  C5⋊D16  D8.D5  C5⋊SD32  C5⋊Q32  D4⋊Dic5  C15⋊D8  C5⋊D24  D4⋊D15  D4⋊D25  C522D8  C5⋊D40  C527D8

Matrix representation of D4⋊D5 in GL4(𝔽41) generated by

1000
0100
00139
00140
,
1000
0100
00024
00120
,
34100
40000
0010
0001
,
13400
04000
0010
00140
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,39,40],[1,0,0,0,0,1,0,0,0,0,0,12,0,0,24,0],[34,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,34,40,0,0,0,0,1,1,0,0,0,40] >;

D4⋊D5 in GAP, Magma, Sage, TeX

D_4\rtimes D_5
% in TeX

G:=Group("D4:D5");
// GroupNames label

G:=SmallGroup(80,15);
// by ID

G=gap.SmallGroup(80,15);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,61,182,97,42,1604]);
// Polycyclic

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

Export

Subgroup lattice of D4⋊D5 in TeX
Character table of D4⋊D5 in TeX

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