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G = D4×F5order 160 = 25·5

Direct product of D4 and F5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4×F5, D203C4, D10.12C23, C5⋊(C4×D4), C20⋊(C2×C4), C5⋊D4⋊C4, D10⋊(C2×C4), C4⋊F52C2, C41(C2×F5), (C5×D4)⋊3C4, Dic5⋊(C2×C4), (C4×F5)⋊3C2, (D4×D5).3C2, D5.2(C2×D4), C22⋊F53C2, C221(C2×F5), (C22×F5)⋊1C2, C2.9(C22×F5), D5.2(C4○D4), C10.8(C22×C4), (C2×F5).3C22, (C4×D5).12C22, (C22×D5).16C22, (C2×C10)⋊(C2×C4), Aut(D20), Hol(C20), SmallGroup(160,207)

Series: Derived Chief Lower central Upper central

C1C10 — D4×F5
C1C5D5D10C2×F5C22×F5 — D4×F5
C5C10 — D4×F5
C1C2D4

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, bd=db, dcd-1=c3 >

Subgroups: 332 in 94 conjugacy classes, 38 normal (20 characteristic)
C1, C2, C2 [×6], C4, C4 [×6], C22 [×2], C22 [×7], C5, C2×C4 [×9], D4, D4 [×3], C23 [×2], D5 [×2], D5 [×2], C10, C10 [×2], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5, C20, F5 [×2], F5 [×3], D10, D10 [×2], D10 [×4], C2×C10 [×2], C4×D4, C4×D5, D20, C5⋊D4 [×2], C5×D4, C2×F5 [×2], C2×F5 [×2], C2×F5 [×4], C22×D5 [×2], C4×F5, C4⋊F5, C22⋊F5 [×2], D4×D5, C22×F5 [×2], D4×F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×F5 [×3], C22×F5, D4×F5

Character table of D4×F5

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L510A10B10C20
 size 1122551010255551010101010101044888
ρ11111111111111111111111111    trivial
ρ211-1-111-1-11111111-1-1-1-1111-1-11    linear of order 2
ρ311-1111-11-1-1-1-1-11-111-1-1111-11-1    linear of order 2
ρ4111-1111-1-1-1-1-1-11-1-1-1111111-1-1    linear of order 2
ρ511-1111-11-11111-1-1-1-111-111-11-1    linear of order 2
ρ6111-1111-1-11111-1-111-1-1-1111-1-1    linear of order 2
ρ7111111111-1-1-1-1-11-1-1-1-1-111111    linear of order 2
ρ811-1-111-1-11-1-1-1-1-111111-111-1-11    linear of order 2
ρ9111-1-1-1-11-1-iii-ii1i-i-ii-i111-1-1    linear of order 4
ρ1011-1-1-1-1111-iii-i-i-1-ii-iii11-1-11    linear of order 4
ρ1111-1-1-1-1111i-i-iii-1i-ii-i-i11-1-11    linear of order 4
ρ12111-1-1-1-11-1i-i-ii-i1-iii-ii111-1-1    linear of order 4
ρ1311-11-1-11-1-1-iii-ii1-iii-i-i11-11-1    linear of order 4
ρ141111-1-1-1-11-iii-i-i-1i-ii-ii11111    linear of order 4
ρ151111-1-1-1-11i-i-iii-1-ii-ii-i11111    linear of order 4
ρ1611-11-1-11-1-1i-i-ii-i1i-i-iii11-11-1    linear of order 4
ρ172-2002-20002-22-200000002-2000    orthogonal lifted from D4
ρ182-2002-2000-22-2200000002-2000    orthogonal lifted from D4
ρ192-200-220002i2i-2i-2i00000002-2000    complex lifted from C4○D4
ρ202-200-22000-2i-2i2i2i00000002-2000    complex lifted from C4○D4
ρ21444-40000-400000000000-1-1-111    orthogonal lifted from C2×F5
ρ2244-440000-400000000000-1-11-11    orthogonal lifted from C2×F5
ρ2344-4-40000400000000000-1-111-1    orthogonal lifted from C2×F5
ρ2444440000400000000000-1-1-1-1-1    orthogonal lifted from F5
ρ258-8000000000000000000-22000    orthogonal faithful

Permutation representations of D4×F5
On 20 points - transitive group 20T42
Generators in S20
(1 11 6 16)(2 12 7 17)(3 13 8 18)(4 14 9 19)(5 15 10 20)
(11 16)(12 17)(13 18)(14 19)(15 20)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 6)(2 8 5 9)(3 10 4 7)(11 16)(12 18 15 19)(13 20 14 17)

G:=sub<Sym(20)| (1,11,6,16)(2,12,7,17)(3,13,8,18)(4,14,9,19)(5,15,10,20), (11,16)(12,17)(13,18)(14,19)(15,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,6)(2,8,5,9)(3,10,4,7)(11,16)(12,18,15,19)(13,20,14,17)>;

G:=Group( (1,11,6,16)(2,12,7,17)(3,13,8,18)(4,14,9,19)(5,15,10,20), (11,16)(12,17)(13,18)(14,19)(15,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,6)(2,8,5,9)(3,10,4,7)(11,16)(12,18,15,19)(13,20,14,17) );

G=PermutationGroup([(1,11,6,16),(2,12,7,17),(3,13,8,18),(4,14,9,19),(5,15,10,20)], [(11,16),(12,17),(13,18),(14,19),(15,20)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,6),(2,8,5,9),(3,10,4,7),(11,16),(12,18,15,19),(13,20,14,17)])

G:=TransitiveGroup(20,42);

D4×F5 is a maximal subgroup of
D40⋊C4  SD16⋊F5  D10.C24  D5.2+ 1+4  D603C4  C3⋊D4⋊F5
D4×F5 is a maximal quotient of
C5⋊C88D4  C5⋊C8⋊D4  D10⋊M4(2)  Dic5⋊M4(2)  D10⋊(C4⋊C4)  C10.(C4×D4)  D202C8  D102M4(2)  C20⋊M4(2)  C4⋊C45F5  C20⋊(C4⋊C4)  D40⋊C4  D85F5  D8⋊F5  SD16⋊F5  SD163F5  SD162F5  Dic20⋊C4  Q165F5  Q16⋊F5  C5⋊C87D4  C202M4(2)  (C2×F5)⋊D4  C2.(D4×F5)  D603C4  C3⋊D4⋊F5

Matrix representation of D4×F5 in GL6(𝔽41)

40390000
110000
0040000
0004000
0000400
0000040
,
4000000
110000
0040000
0004000
0000400
0000040
,
100000
010000
0000040
0010040
0001040
0000140
,
3200000
0320000
000010
001000
000001
000100

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

D4×F5 in GAP, Magma, Sage, TeX

D_4\times F_5
% in TeX

G:=Group("D4xF5");
// GroupNames label

G:=SmallGroup(160,207);
// by ID

G=gap.SmallGroup(160,207);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,188,2309,599]);
// 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,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

Export

Character table of D4×F5 in TeX

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