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G = D42D5order 80 = 24·5

The semidirect product of D4 and D5 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D42D5, D4Dic5, C4.5D10, Dic103C2, C10.6C23, C20.5C22, C22.1D10, D10.2C22, Dic5.8C22, (C4×D5)⋊2C2, (C5×D4)⋊3C2, C52(C4○D4), C5⋊D42C2, (C2×C10).C22, (C2×Dic5)⋊3C2, C2.7(C22×D5), SmallGroup(80,40)

Series: Derived Chief Lower central Upper central

C1C10 — D42D5
C1C5C10D10C4×D5 — D42D5
C5C10 — D42D5
C1C2D4

Generators and relations for D42D5
 G = < a,b,c,d | a4=b2=c5=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >

2C2
2C2
10C2
5C4
5C22
5C4
5C4
2C10
2C10
2D5
5C2×C4
5D4
5D4
5C2×C4
5Q8
5C2×C4
5C4○D4

Character table of D42D5

 class 12A2B2C2D4A4B4C4D4E5A5B10A10B10C10D10E10F20A20B
 size 11221025510102222444444
ρ111111111111111111111    trivial
ρ2111-1-1-111-1111111-1-11-1-1    linear of order 2
ρ311-1-11111-1-11111-1-1-1-111    linear of order 2
ρ411-11-1-1111-11111-111-1-1-1    linear of order 2
ρ51111-11-1-1-1-11111111111    linear of order 2
ρ6111-11-1-1-11-111111-1-11-1-1    linear of order 2
ρ711-111-1-1-1-111111-111-1-1-1    linear of order 2
ρ811-1-1-11-1-1111111-1-1-1-111    linear of order 2
ρ92222020000-1+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
ρ102222020000-1-5/2-1+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
ρ1122-2-2020000-1-5/2-1+5/2-1+5/2-1-5/21-5/21+5/21-5/21+5/2-1+5/2-1-5/2    orthogonal lifted from D10
ρ1222-2-2020000-1+5/2-1-5/2-1-5/2-1+5/21+5/21-5/21+5/21-5/2-1-5/2-1+5/2    orthogonal lifted from D10
ρ1322-220-20000-1-5/2-1+5/2-1+5/2-1-5/21-5/2-1-5/2-1+5/21+5/21-5/21+5/2    orthogonal lifted from D10
ρ14222-20-20000-1-5/2-1+5/2-1+5/2-1-5/2-1+5/21+5/21-5/2-1-5/21-5/21+5/2    orthogonal lifted from D10
ρ15222-20-20000-1+5/2-1-5/2-1-5/2-1+5/2-1-5/21-5/21+5/2-1+5/21+5/21-5/2    orthogonal lifted from D10
ρ1622-220-20000-1+5/2-1-5/2-1-5/2-1+5/21+5/2-1+5/2-1-5/21-5/21+5/21-5/2    orthogonal lifted from D10
ρ172-200002i-2i0022-2-2000000    complex lifted from C4○D4
ρ182-20000-2i2i0022-2-2000000    complex lifted from C4○D4
ρ194-400000000-1+5-1-51+51-5000000    symplectic faithful, Schur index 2
ρ204-400000000-1-5-1+51-51+5000000    symplectic faithful, Schur index 2

Smallest permutation representation of D42D5
On 40 points
Generators in S40
(1 29 9 24)(2 30 10 25)(3 26 6 21)(4 27 7 22)(5 28 8 23)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)
(1 34)(2 35)(3 31)(4 32)(5 33)(6 36)(7 37)(8 38)(9 39)(10 40)(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)(22 25)(23 24)(27 30)(28 29)(31 36)(32 40)(33 39)(34 38)(35 37)

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

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

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

D42D5 is a maximal subgroup of
D4⋊F5  D8⋊D5  D83D5  SD16⋊D5  SD163D5  D4.F5  D46D10  D5×C4○D4  D4.10D10  D12⋊D5  D125D5  C30.C23  Dic3.D10  D42D15  D42D25  D205D5  D20⋊D5  Dic5.D10  D10.4D10  C20.D10  D4.A5
D42D5 is a maximal quotient of
C23.11D10  Dic5.14D4  C23.D10  Dic54D4  D10.12D4  Dic5.5D4  C22.D20  Dic53Q8  Dic5.Q8  C4.Dic10  C4⋊C47D5  D102Q8  C4⋊C4⋊D5  D4×Dic5  C23.18D10  C20.17D4  C202D4  Dic5⋊D4  D12⋊D5  D125D5  C30.C23  Dic3.D10  D42D15  D42D25  D205D5  D20⋊D5  Dic5.D10  D10.4D10  C20.D10

Matrix representation of D42D5 in GL4(𝔽41) generated by

1000
0100
0090
002532
,
1000
0100
003236
00169
,
0100
40600
0010
0001
,
0100
1000
0010
002140
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,9,25,0,0,0,32],[1,0,0,0,0,1,0,0,0,0,32,16,0,0,36,9],[0,40,0,0,1,6,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,21,0,0,0,40] >;

D42D5 in GAP, Magma, Sage, TeX

D_4\rtimes_2D_5
% in TeX

G:=Group("D4:2D5");
// GroupNames label

G:=SmallGroup(80,40);
// by ID

G=gap.SmallGroup(80,40);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,46,182,97,1604]);
// Polycyclic

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

Export

Subgroup lattice of D42D5 in TeX
Character table of D42D5 in TeX

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