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G = Q8⋊F5  order 160 = 25·5

1st semidirect product of Q8 and F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q8⋊1F5, D5.2Q16, Dic10⋊2C4, D10.19D4, Dic5.3D4, D5.3SD16, C5⋊(Q8⋊C4), (C5×Q8)⋊1C4, C4⋊F5.1C2, C4.3(C2×F5), C20.3(C2×C4), D5⋊C8.1C2, (Q8×D5).2C2, (C4×D5).9C22, C2.8(C22⋊F5), C10.7(C22⋊C4), SmallGroup(160,84)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C20 — Q8⋊F5
Chief series
C1 — C5 — C10 — D10 — C4×D5 — C4⋊F5 — Q8⋊F5
Lower central
C5 — C10 — C20 — Q8⋊F5
Upper central
C1 — C2 — C4 — Q8

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

C1
C2
5C2
5C2
C5
C4
2C4
5C22
5C4
10C4
20C4
D5
C10
D5
Q8
5Q8
5C2×C4
10C2×C4
10C2×C4
10C8
10Q8
C20
Dic5
D10
2C20
2Dic5
4F5
5C2×Q8
5C4⋊C4
5C2×C8
Dic10
C5×Q8
C4×D5
2Dic10
2C5⋊C8
2C2×F5
2C4×D5
5Q8⋊C4
Q8×D5
C4⋊F5
D5⋊C8
Q8⋊F5

Character table of Q8⋊F5

 class 12A2B2C4A4B4C4D4E4F58A8B8C8D1020A20B20C
 size 115524102020204101010104888
ρ11111111111111111111    trivial
ρ211111-111-111-1-1-1-11-1-11    linear of order 2
ρ311111-11-1-1-1111111-1-11    linear of order 2
ρ41111111-11-11-1-1-1-11111    linear of order 2
ρ511-1-11-1-1-i1i1i-i-ii1-1-11    linear of order 4
ρ611-1-111-1-i-1i1-iii-i1111    linear of order 4
ρ711-1-11-1-1i1-i1-iii-i1-1-11    linear of order 4
ρ811-1-111-1i-1-i1i-i-ii1111    linear of order 4
ρ92222-20-200020000200-2    orthogonal lifted from D4
ρ1022-2-2-20200020000200-2    orthogonal lifted from D4
ρ112-2-220000002-√2-√2√2√2-2000    symplectic lifted from Q16, Schur index 2
ρ122-2-220000002√2√2-√2-√2-2000    symplectic lifted from Q16, Schur index 2
ρ132-22-20000002√-2-√-2√-2-√-2-2000    complex lifted from SD16
ρ142-22-20000002-√-2√-2-√-2√-2-2000    complex lifted from SD16
ρ154400440000-10000-1-1-1-1    orthogonal lifted from F5
ρ1644004-40000-10000-111-1    orthogonal lifted from C2×F5
ρ174400-400000-10000-1√5-√51    orthogonal lifted from C22⋊F5
ρ184400-400000-10000-1-√5√51    orthogonal lifted from C22⋊F5
ρ198-800000000-200002000    symplectic faithful, Schur index 2

Smallest permutation representation of Q8⋊F5
►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)(11 16)(12 18 15 19)(13 20 14 17)(21 31)(22 33 25 34)(23 35 24 32)(26 36)(27 38 30 39)(28 40 29 37)
 
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)(11,16)(12,18,15,19)(13,20,14,17)(21,31)(22,33,25,34)(23,35,24,32)(26,36)(27,38,30,39)(28,40,29,37)>;
 
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)(11,16)(12,18,15,19)(13,20,14,17)(21,31)(22,33,25,34)(23,35,24,32)(26,36)(27,38,30,39)(28,40,29,37) );
 
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),(11,16),(12,18,15,19),(13,20,14,17),(21,31),(22,33,25,34),(23,35,24,32),(26,36),(27,38,30,39),(28,40,29,37)]])
 

Q8⋊F5 is a maximal subgroup of
 SD16×F5  SD16⋊F5  Q16×F5  Dic20⋊C4  (C2×Q8)⋊4F5  C4○D4⋊F5  C4○D20⋊C4  Dic6⋊F5  Dic30⋊C4  Dic10⋊2Dic3  D10.S4
Q8⋊F5 is a maximal quotient of
 D10.1Q16  Dic10⋊1C8  Dic5.D8  D10.18D8  D10.Q16  Dic5.12Q16  Dic5.Q16  Dic6⋊F5  Dic30⋊C4  Dic10⋊2Dic3

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

40390000
110000
001000
000100
000010
000001
,
0110000
2600000
001000
000100
000010
000001
,
100000
010000
0000040
0010040
0001040
0000140
,
3200000
990000
000010
001000
000001
000100

G:=sub<GL(6,GF(41))| [40,1,0,0,0,0,39,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,26,0,0,0,0,11,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[32,9,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;
 

Q8⋊F5 in GAP, Magma, Sage, TeX 

Q_8\rtimes F_5
 
% in TeX
 
G:=Group("Q8:F5");
 
// GroupNames label
 
G:=SmallGroup(160,84);
 
// by ID
 
G=gap.SmallGroup(160,84);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,103,579,297,69,2309,1169]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=c^5=d^4=1,b^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*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⋊F5 in TeX
Character table of Q8⋊F5 in TeX

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