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G = Q8⋊2F5  order 160 = 25·5

2nd semidirect product of Q8 and F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q8⋊2F5, D20⋊2C4, D10.3D4, Dic5.22D4, C5⋊2C4≀C2, (C4×F5)⋊2C2, (C5×Q8)⋊2C4, C4.F5⋊2C2, C4.4(C2×F5), C20.4(C2×C4), Q8⋊2D5.2C2, C2.9(C22⋊F5), C10.8(C22⋊C4), (C4×D5).10C22, SmallGroup(160,85)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C20 — Q8⋊2F5
Chief series
C1 — C5 — C10 — Dic5 — C4×D5 — C4.F5 — Q8⋊2F5
Lower central
C5 — C10 — C20 — Q8⋊2F5
Upper central
C1 — C2 — C4 — Q8

Generators and relations for Q8⋊2F5
 G = < a,b,c,d | a4=c5=d4=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a-1b, dcd-1=c3 >

C1
C2
10C2
20C2
C5
C4
2C4
5C22
5C4
10C4
10C22
10C4
C10
2D5
4D5
Q8
5D4
5C2×C4
10D4
10C2×C4
10C2×C4
10C8
Dic5
D10
C20
2C20
2F5
2D10
2F5
5C4○D4
5C42
5M4(2)
D20
C4×D5
C5×Q8
2C5⋊C8
2C4×D5
2D20
2C2×F5
5C4≀C2
C4.F5
C4×F5
Q8⋊2D5
Q8⋊2F5

Character table of Q8⋊2F5

 class 12A2B2C4A4B4C4D4E4F4G4H58A8B1020A20B20C
 size 111020245510101010420204888
ρ11111111111111111111    trivial
ρ2111-11-111-1-1-1-111111-1-1    linear of order 2
ρ3111-11-11111111-1-111-1-1    linear of order 2
ρ411111111-1-1-1-11-1-11111    linear of order 2
ρ511-111-1-1-1i-ii-i1i-i11-1-1    linear of order 4
ρ611-111-1-1-1-ii-ii1-ii11-1-1    linear of order 4
ρ711-1-111-1-1i-ii-i1-ii1111    linear of order 4
ρ811-1-111-1-1-ii-ii1i-i1111    linear of order 4
ρ922-20-202200002002-200    orthogonal lifted from D4
ρ102220-20-2-200002002-200    orthogonal lifted from D4
ρ112-20000-2i2i-1+i-1-i1-i1+i200-2000    complex lifted from C4≀C2
ρ122-200002i-2i-1-i-1+i1+i1-i200-2000    complex lifted from C4≀C2
ρ132-200002i-2i1+i1-i-1-i-1+i200-2000    complex lifted from C4≀C2
ρ142-20000-2i2i1-i1+i-1+i-1-i200-2000    complex lifted from C4≀C2
ρ1544004-4000000-100-1-111    orthogonal lifted from C2×F5
ρ16440044000000-100-1-1-1-1    orthogonal lifted from F5
ρ174400-40000000-100-11-√5√5    orthogonal lifted from C22⋊F5
ρ184400-40000000-100-11√5-√5    orthogonal lifted from C22⋊F5
ρ198-80000000000-2002000    orthogonal faithful

Smallest permutation representation of Q8⋊2F5
►On 40 points
Generators in S40
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)

(1 26 6 21)(2 27 7 22)(3 28 8 23)(4 29 9 24)(5 30 10 25)(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)

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

Q8⋊2F5 is a maximal subgroup of
 SD16⋊3F5  SD16⋊2F5  Q16⋊5F5  Q16⋊F5  (C2×Q8)⋊6F5  D5⋊C4≀C2  D4⋊F5⋊C2  D60⋊2C4  D60⋊5C4  D20⋊2Dic3  C5⋊U2(𝔽3)
Q8⋊2F5 is a maximal quotient of
 D10.1D8  C10.C4≀C2  D20⋊C8  C20.C42  D10.Q16  (C2×Q8).F5  Dic5.12Q16  D60⋊2C4  D60⋊5C4  D20⋊2Dic3

Matrix representation of Q8⋊2F5 ►in GL6(𝔽41)

900000
0320000
0040000
0004000
0000400
0000040
,
010000
4000000
0022033
003819380
000381938
0033022
,
100000
010000
0040404040
001000
000100
000010
,
100000
090000
001000
000001
000100
0040404040

G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,22,38,0,3,0,0,0,19,38,3,0,0,3,38,19,0,0,0,3,0,38,22],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,1,0,0,40,0,0,0],[1,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,1,0,40] >;
 

Q8⋊2F5 in GAP, Magma, Sage, TeX 

Q_8\rtimes_2F_5
 
% in TeX
 
G:=Group("Q8:2F5");
 
// GroupNames label
 
G:=SmallGroup(160,85);
 
// by ID
 
G=gap.SmallGroup(160,85);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,103,86,579,297,69,2309,1169]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=c^5=d^4=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^3>;
 
// generators/relations
 

Export

Subgroup lattice of Q8⋊2F5 in TeX
Character table of Q8⋊2F5 in TeX

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