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G = D10⋊F5  order 400 = 24·52

2nd semidirect product of D10 and F5 acting via F5/D5=C2

metabelian, supersoluble, monomial

Aliases: D10⋊2F5, C5⋊D5.1D4, (D5×C10)⋊2C4, C10.2(C2×F5), C2.3(D5⋊F5), C5⋊1(C22⋊F5), C52⋊2(C22⋊C4), (C2×D52).2C2, (C5×C10).9(C2×C4), (C2×C52⋊C4)⋊1C2, (C2×C5⋊F5)⋊1C2, (C2×C5⋊D5).4C22, SmallGroup(400,125)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5×C10 — D10⋊F5
Chief series
C1 — C5 — C52 — C5⋊D5 — C2×C5⋊D5 — C2×C5⋊F5 — D10⋊F5
Lower central
C52 — C5×C10 — D10⋊F5
Upper central
C1 — C2

Generators and relations for D10⋊F5
 G = < a,b,c,d | a10=b2=c5=d4=1, bab=a-1, ac=ca, dad-1=a3, bc=cb, dbd-1=a7b, dcd-1=c3 >

Subgroups: 748 in 84 conjugacy classes, 19 normal (11 characteristic)
C1, C2, C2, C4, C22, C5, C5, C2×C4, C23, D5, C10, C10, C22⋊C4, F5, D10, D10, C2×C10, C52, C2×F5, C22×D5, C5×D5, C5⋊D5, C5×C10, C22⋊F5, C5⋊F5, C52⋊C4, D52, D5×C10, C2×C5⋊D5, C2×C5⋊F5, C2×C52⋊C4, C2×D52, D10⋊F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, F5, C2×F5, C22⋊F5, D5⋊F5, D10⋊F5

Character table of D10⋊F5

 class 12A2B2C2D2E4A4B4C4D5A5B5C5D10A10B10C10D10E10F10G10H
 size 1110102525505050504488448820202020
ρ11111111111111111111111    trivial
ρ211-1-111-111-111111111-1-1-1-1    linear of order 2
ρ3111111-1-1-1-1111111111111    linear of order 2
ρ411-1-1111-1-1111111111-1-1-1-1    linear of order 2
ρ511-11-1-1-ii-ii111111111-11-1    linear of order 4
ρ6111-1-1-1ii-i-i11111111-11-11    linear of order 4
ρ711-11-1-1i-ii-i111111111-11-1    linear of order 4
ρ8111-1-1-1-i-iii11111111-11-11    linear of order 4
ρ92-2002-200002222-2-2-2-20000    orthogonal lifted from D4
ρ102-200-2200002222-2-2-2-20000    orthogonal lifted from D4
ρ1144400000004-1-1-14-1-1-10-10-1    orthogonal lifted from F5
ρ1244-400000004-1-1-14-1-1-10101    orthogonal lifted from C2×F5
ρ13440-4000000-14-1-1-14-1-11010    orthogonal lifted from C2×F5
ρ144404000000-14-1-1-14-1-1-10-10    orthogonal lifted from F5
ρ154-4000000004-1-1-1-41110√50-√5    orthogonal lifted from C22⋊F5
ρ164-400000000-14-1-11-411-√50√50    orthogonal lifted from C22⋊F5
ρ174-400000000-14-1-11-411√50-√50    orthogonal lifted from C22⋊F5
ρ184-4000000004-1-1-1-41110-√50√5    orthogonal lifted from C22⋊F5
ρ198-800000000-2-23-2222-30000    orthogonal faithful
ρ208-800000000-2-2-2322-320000    orthogonal faithful
ρ218800000000-2-23-2-2-2-230000    orthogonal lifted from D5⋊F5
ρ228800000000-2-2-23-2-23-20000    orthogonal lifted from D5⋊F5

Permutation representations of D10⋊F5
►On 20 points - transitive group 20T101
Generators in S20
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)

(1 12)(2 11)(3 20)(4 19)(5 18)(6 17)(7 16)(8 15)(9 14)(10 13)

(1 5 9 3 7)(2 6 10 4 8)(11 17 13 19 15)(12 18 14 20 16)

(2 8 10 4)(3 5 9 7)(11 14 15 12)(13 18)(16 19 20 17)
 
G:=sub<Sym(20)| (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,12)(2,11)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13), (1,5,9,3,7)(2,6,10,4,8)(11,17,13,19,15)(12,18,14,20,16), (2,8,10,4)(3,5,9,7)(11,14,15,12)(13,18)(16,19,20,17)>;
 
G:=Group( (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,12)(2,11)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13), (1,5,9,3,7)(2,6,10,4,8)(11,17,13,19,15)(12,18,14,20,16), (2,8,10,4)(3,5,9,7)(11,14,15,12)(13,18)(16,19,20,17) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20)], [(1,12),(2,11),(3,20),(4,19),(5,18),(6,17),(7,16),(8,15),(9,14),(10,13)], [(1,5,9,3,7),(2,6,10,4,8),(11,17,13,19,15),(12,18,14,20,16)], [(2,8,10,4),(3,5,9,7),(11,14,15,12),(13,18),(16,19,20,17)]])
 
G:=TransitiveGroup(20,101);
 

Matrix representation of D10⋊F5 ►in GL8(ℤ)

000-10000
11110000
-10000000
0-1000000
0000000-1
00001111
0000-1000
00000-100
,
0000000-1
000000-10
00000-100
0000-1000
000-10000
00-100000
0-1000000
-10000000
,
01000000
00100000
00010000
-1-1-1-10000
0000-1-1-1-1
00001000
00000100
00000010
,
10000000
00010000
01000000
-1-1-1-10000
0000-1000
0000000-1
00000-100
00001111

G:=sub<GL(8,Integers())| [0,1,-1,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,-1,1,0,0],[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,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0],[0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,0,-1,0,0,0],[1,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,-1,0,1] >;
 

D10⋊F5 in GAP, Magma, Sage, TeX 

D_{10}\rtimes F_5
 
% in TeX
 
G:=Group("D10:F5");
 
// GroupNames label
 
G:=SmallGroup(400,125);
 
// by ID
 
G=gap.SmallGroup(400,125);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,1444,970,496,8645,2897]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^10=b^2=c^5=d^4=1,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^3,b*c=c*b,d*b*d^-1=a^7*b,d*c*d^-1=c^3>;
 
// generators/relations
 

Export

Character table of D10⋊F5 in TeX

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