Copied to
clipboard

G = Q8⋊2D5  order 80 = 24·5

The semidirect product of Q8 and D5 acting through Inn(Q8)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q8⋊2D5, Q8○Dic5, D20⋊4C2, C4.7D10, C20.7C22, C10.8C23, D10.3C22, Dic5.9C22, (C4×D5)⋊3C2, C5⋊3(C4○D4), (C5×Q8)⋊3C2, C2.9(C22×D5), SmallGroup(80,42)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — Q8⋊2D5
Chief series
C1 — C5 — C10 — D10 — C4×D5 — Q8⋊2D5
Lower central
C5 — C10 — Q8⋊2D5
Upper central
C1 — C2 — Q8

Generators and relations for Q8⋊2D5
 G = < a,b,c,d | a4=c5=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >

C1
C2
10C2
10C2
10C2
C5
C4
C4
C4
5C22
5C22
5C22
5C4
C10
2D5
2D5
2D5
Q8
5C2×C4
5D4
5D4
5C2×C4
5D4
5C2×C4
D10
D10
Dic5
C20
D10
C20
C20
5C4○D4
D20
C4×D5
C4×D5
D20
D20
C4×D5
C5×Q8
Q8⋊2D5

Character table of Q8⋊2D5

 class 12A2B2C2D4A4B4C4D4E5A5B10A10B20A20B20C20D20E20F
 size 11101010222552222444444
ρ111111111111111111111    trivial
ρ211-111-11-1-1-11111-11-1-11-1    linear of order 2
ρ3111-111-1-1-1-111111-1-1-1-11    linear of order 2
ρ411-1-11-1-11111111-1-111-1-1    linear of order 2
ρ51111-1-1-11-1-11111-1-111-1-1    linear of order 2
ρ611-11-11-1-11111111-1-1-1-11    linear of order 2
ρ7111-1-1-11-1111111-11-1-11-1    linear of order 2
ρ811-1-1-1111-1-11111111111    linear of order 2
ρ922000-22-200-1-√5/2-1+√5/2-1+√5/2-1-√5/21-√5/2-1-√5/21+√5/21-√5/2-1+√5/21+√5/2    orthogonal lifted from D10
ρ1022000-22-200-1+√5/2-1-√5/2-1-√5/2-1+√5/21+√5/2-1+√5/21-√5/21+√5/2-1-√5/21-√5/2    orthogonal lifted from D10
ρ11220002-2-200-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1+√5/21+√5/21+√5/21-√5/21-√5/2-1-√5/2    orthogonal lifted from D10
ρ12220002-2-200-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1-√5/21-√5/21-√5/21+√5/21+√5/2-1+√5/2    orthogonal lifted from D10
ρ132200022200-1-√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
ρ1422000-2-2200-1+√5/2-1-√5/2-1-√5/2-1+√5/21+√5/21-√5/2-1+√5/2-1-√5/21+√5/21-√5/2    orthogonal lifted from D10
ρ1522000-2-2200-1-√5/2-1+√5/2-1+√5/2-1-√5/21-√5/21+√5/2-1-√5/2-1+√5/21-√5/21+√5/2    orthogonal lifted from D10
ρ162200022200-1+√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
ρ172-2000000-2i2i22-2-2000000    complex lifted from C4○D4
ρ182-20000002i-2i22-2-2000000    complex lifted from C4○D4
ρ194-400000000-1+√5-1-√51+√51-√5000000    orthogonal faithful
ρ204-400000000-1-√5-1+√51-√51+√5000000    orthogonal faithful

Smallest permutation representation of Q8⋊2D5
►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 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 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)| (1,19,9,14)(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), (1,29,9,24)(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), (1,2,3,4,5)(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( (1,19,9,14)(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), (1,29,9,24)(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), (1,2,3,4,5)(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([[(1,19,9,14),(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)], [(1,29,9,24),(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)], [(1,2,3,4,5),(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)]])
 

Q8⋊2D5 is a maximal subgroup of
 Q8⋊2F5  D40⋊C2  SD16⋊3D5  Q16⋊D5  Q8.D10  Q8.F5  Q8.10D10  D5×C4○D4  D4⋊8D10  Dic5.A4  D20⋊S3  C12.28D10  Q8⋊3D15  Q8⋊2D25  D20⋊D5  Dic10⋊5D5  C20.26D10  Q8.A5
Q8⋊2D5 is a maximal quotient of
 C4.Dic10  C4⋊C4⋊7D5  D20⋊8C4  D10.13D4  C4⋊D20  C4⋊C4⋊D5  Q8×Dic5  D10⋊3Q8  C20.23D4  D20⋊S3  C12.28D10  Q8⋊3D15  Q8⋊2D25  D20⋊D5  Dic10⋊5D5  C20.26D10

Matrix representation of Q8⋊2D5 ►in GL4(𝔽41) generated by

40000
04000
0001
00400
,
40000
04000
0090
00032
,
0100
40600
0010
0001
,
0100
1000
0010
00040
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,0,40,0,0,1,0],[40,0,0,0,0,40,0,0,0,0,9,0,0,0,0,32],[0,40,0,0,1,6,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,40] >;
 

Q8⋊2D5 in GAP, Magma, Sage, TeX 

Q_8\rtimes_2D_5
 
% in TeX
 
G:=Group("Q8:2D5");
 
// GroupNames label
 
G:=SmallGroup(80,42);
 
// by ID
 
G=gap.SmallGroup(80,42);
 
# by ID
 
G:=PCGroup([5,-2,-2,-2,-2,-5,46,182,97,42,1604]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=c^5=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of Q8⋊2D5 in TeX
Character table of Q8⋊2D5 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁