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G = D4⋊F5order 160 = 25·5

2nd semidirect product of D4 and F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D42F5, D10.2D4, Dic101C4, Dic5.21D4, C51C4≀C2, (C5×D4)⋊2C4, (C4×F5)⋊1C2, C4.F51C2, C4.2(C2×F5), C20.2(C2×C4), D42D5.2C2, (C4×D5).8C22, C2.7(C22⋊F5), C10.6(C22⋊C4), SmallGroup(160,83)

Series: Derived Chief Lower central Upper central

C1C20 — D4⋊F5
C1C5C10Dic5C4×D5C4.F5 — D4⋊F5
C5C10C20 — D4⋊F5
C1C2C4D4

Generators and relations for D4⋊F5
 G = < a,b,c,d | a4=b2=c5=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a-1b, dcd-1=c3 >

4C2
10C2
2C22
5C4
5C22
10C4
10C4
10C4
2D5
4C10
5Q8
5C2×C4
10D4
10C2×C4
10C2×C4
10C8
2C2×C10
2F5
2Dic5
2F5
5C4○D4
5C42
5M4(2)
2C5⋊C8
2C2×Dic5
2C5⋊D4
2C2×F5
5C4≀C2

Character table of D4⋊F5

 class 12A2B2C4A4B4C4D4E4F4G4H58A8B10A10B10C20
 size 114102551010101020420204888
ρ11111111111111111111    trivial
ρ211-111111111-11-1-11-1-11    linear of order 2
ρ31111111-1-1-1-111-1-11111    linear of order 2
ρ411-11111-1-1-1-1-11111-1-11    linear of order 2
ρ5111-11-1-1-iii-i-11-ii1111    linear of order 4
ρ611-1-11-1-1-iii-i11i-i1-1-11    linear of order 4
ρ7111-11-1-1i-i-ii-11i-i1111    linear of order 4
ρ811-1-11-1-1i-i-ii11-ii1-1-11    linear of order 4
ρ9220-2-22200000200200-2    orthogonal lifted from D4
ρ102202-2-2-200000200200-2    orthogonal lifted from D4
ρ112-20002i-2i-1+i-1-i1+i1-i0200-2000    complex lifted from C4≀C2
ρ122-2000-2i2i-1-i-1+i1-i1+i0200-2000    complex lifted from C4≀C2
ρ132-20002i-2i1-i1+i-1-i-1+i0200-2000    complex lifted from C4≀C2
ρ142-2000-2i2i1+i1-i-1+i-1-i0200-2000    complex lifted from C4≀C2
ρ15444040000000-100-1-1-1-1    orthogonal lifted from F5
ρ1644-4040000000-100-111-1    orthogonal lifted from C2×F5
ρ174400-40000000-100-15-51    orthogonal lifted from C22⋊F5
ρ184400-40000000-100-1-551    orthogonal lifted from C22⋊F5
ρ198-80000000000-2002000    symplectic faithful, Schur index 2

Smallest permutation representation of D4⋊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 36)(2 37)(3 38)(4 39)(5 40)(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)
(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), (1,36)(2,37)(3,38)(4,39)(5,40)(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), (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), (1,36)(2,37)(3,38)(4,39)(5,40)(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), (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)], [(1,36),(2,37),(3,38),(4,39),(5,40),(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)], [(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)])

D4⋊F5 is a maximal subgroup of
D85F5  D8⋊F5  SD163F5  SD162F5  (C2×D4)⋊6F5  D5⋊C4≀C2  D4⋊F5⋊C2  D124F5  D122F5  Dic10⋊Dic3
D4⋊F5 is a maximal quotient of
D10.1Q16  C10.C4≀C2  Dic101C8  C20.C42  D10.SD16  (C2×D4).F5  Dic5.23D8  D124F5  D122F5  Dic10⋊Dic3

Matrix representation of D4⋊F5 in GL6(𝔽41)

3280000
090000
001000
000100
000010
000001
,
20130000
4210000
0040000
0004000
0000400
0000040
,
100000
010000
0040404040
001000
000100
000010
,
32400000
0400000
001000
000001
000100
0040404040

G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,8,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[20,4,0,0,0,0,13,21,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[32,0,0,0,0,0,40,40,0,0,0,0,0,0,1,0,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,1,0,40] >;

D4⋊F5 in GAP, Magma, Sage, TeX

D_4\rtimes F_5
% in TeX

G:=Group("D4:F5");
// GroupNames label

G:=SmallGroup(160,83);
// by ID

G=gap.SmallGroup(160,83);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,86,579,297,69,2309,1169]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^5=d^4=1,b*a*b=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 D4⋊F5 in TeX
Character table of D4⋊F5 in TeX

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