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## G = Dic5⋊F5order 400 = 24·52

### 3rd semidirect product of Dic5 and F5 acting via F5/D5=C2

Aliases: Dic53F5, C51(C4⋊F5), C5⋊D5.2D4, C5⋊D5.1Q8, C522(C4⋊C4), C10.3(C2×F5), (C5×Dic5)⋊2C4, C2.4(D5⋊F5), Dic52D5.4C2, (C5×C10).10(C2×C4), (C2×C52⋊C4).2C2, (C2×C5⋊F5).2C2, (C2×C5⋊D5).5C22, SmallGroup(400,126)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — Dic5⋊F5
 Chief series C1 — C5 — C52 — C5⋊D5 — C2×C5⋊D5 — C2×C5⋊F5 — Dic5⋊F5
 Lower central C52 — C5×C10 — Dic5⋊F5
 Upper central C1 — C2

Generators and relations for Dic5⋊F5
G = < a,b,c,d | a10=c5=d4=1, b2=a5, bab-1=a-1, ac=ca, dad-1=a3, bc=cb, dbd-1=a5b, dcd-1=c3 >

Subgroups: 556 in 68 conjugacy classes, 19 normal (11 characteristic)
C1, C2, C2, C4, C22, C5, C5, C2×C4, D5, C10, C10, C4⋊C4, Dic5, C20, F5, D10, C52, C4×D5, C2×F5, C5⋊D5, C5×C10, C4⋊F5, C5×Dic5, C5⋊F5, C52⋊C4, C2×C5⋊D5, Dic52D5, C2×C5⋊F5, C2×C52⋊C4, Dic5⋊F5
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C4⋊C4, F5, C2×F5, C4⋊F5, D5⋊F5, Dic5⋊F5

Character table of Dic5⋊F5

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 10A 10B 10C 10D 20A 20B 20C 20D size 1 1 25 25 10 10 50 50 50 50 4 4 8 8 4 4 8 8 20 20 20 20 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 -1 1 -1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ4 1 1 1 1 -1 -1 -1 1 1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 -1 -1 -1 1 i -i i -i 1 1 1 1 1 1 1 1 1 -1 -1 1 linear of order 4 ρ6 1 1 -1 -1 1 -1 i i -i -i 1 1 1 1 1 1 1 1 -1 1 1 -1 linear of order 4 ρ7 1 1 -1 -1 -1 1 -i i -i i 1 1 1 1 1 1 1 1 1 -1 -1 1 linear of order 4 ρ8 1 1 -1 -1 1 -1 -i -i i i 1 1 1 1 1 1 1 1 -1 1 1 -1 linear of order 4 ρ9 2 -2 -2 2 0 0 0 0 0 0 2 2 2 2 -2 -2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 2 -2 0 0 0 0 0 0 2 2 2 2 -2 -2 -2 -2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ11 4 4 0 0 -4 0 0 0 0 0 -1 4 -1 -1 4 -1 -1 -1 0 1 1 0 orthogonal lifted from C2×F5 ρ12 4 4 0 0 0 -4 0 0 0 0 4 -1 -1 -1 -1 4 -1 -1 1 0 0 1 orthogonal lifted from C2×F5 ρ13 4 4 0 0 4 0 0 0 0 0 -1 4 -1 -1 4 -1 -1 -1 0 -1 -1 0 orthogonal lifted from F5 ρ14 4 4 0 0 0 4 0 0 0 0 4 -1 -1 -1 -1 4 -1 -1 -1 0 0 -1 orthogonal lifted from F5 ρ15 4 -4 0 0 0 0 0 0 0 0 -1 4 -1 -1 -4 1 1 1 0 -√-5 √-5 0 complex lifted from C4⋊F5 ρ16 4 -4 0 0 0 0 0 0 0 0 4 -1 -1 -1 1 -4 1 1 √-5 0 0 -√-5 complex lifted from C4⋊F5 ρ17 4 -4 0 0 0 0 0 0 0 0 4 -1 -1 -1 1 -4 1 1 -√-5 0 0 √-5 complex lifted from C4⋊F5 ρ18 4 -4 0 0 0 0 0 0 0 0 -1 4 -1 -1 -4 1 1 1 0 √-5 -√-5 0 complex lifted from C4⋊F5 ρ19 8 -8 0 0 0 0 0 0 0 0 -2 -2 3 -2 2 2 2 -3 0 0 0 0 orthogonal faithful ρ20 8 -8 0 0 0 0 0 0 0 0 -2 -2 -2 3 2 2 -3 2 0 0 0 0 orthogonal faithful ρ21 8 8 0 0 0 0 0 0 0 0 -2 -2 3 -2 -2 -2 -2 3 0 0 0 0 orthogonal lifted from D5⋊F5 ρ22 8 8 0 0 0 0 0 0 0 0 -2 -2 -2 3 -2 -2 3 -2 0 0 0 0 orthogonal lifted from D5⋊F5

Permutation representations of Dic5⋊F5
On 20 points - transitive group 20T108
Generators in S20
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)
(1 14 6 19)(2 13 7 18)(3 12 8 17)(4 11 9 16)(5 20 10 15)
(1 9 7 5 3)(2 10 8 6 4)(11 13 15 17 19)(12 14 16 18 20)
(2 8 10 4)(3 5 9 7)(11 18 17 20)(12 15 16 13)(14 19)

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

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

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

G:=TransitiveGroup(20,108);

Matrix representation of Dic5⋊F5 in GL8(ℤ)

 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 1 1 1
,
 0 0 0 0 1 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 -1 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1 -1 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1 -1 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 0 1 1 1 1

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

Dic5⋊F5 in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes F_5
% in TeX

G:=Group("Dic5:F5");
// GroupNames label

G:=SmallGroup(400,126);
// by ID

G=gap.SmallGroup(400,126);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,55,1444,970,496,8645,2897]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^5=d^4=1,b^2=a^5,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^3,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^3>;
// generators/relations

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