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G = Dic5⋊F5  order 400 = 24·52

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

metabelian, supersoluble, monomial

Aliases: Dic5⋊3F5, C5⋊1(C4⋊F5), C5⋊D5.2D4, C5⋊D5.1Q8, C52⋊2(C4⋊C4), C10.3(C2×F5), (C5×Dic5)⋊2C4, C2.4(D5⋊F5), Dic5⋊2D5.4C2, (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, Dic5⋊2D5, 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 12A2B2C4A4B4C4D4E4F5A5B5C5D10A10B10C10D20A20B20C20D
 size 1125251010505050504488448820202020
ρ11111111111111111111111    trivial
ρ21111-1-11-1-1111111111-1-1-1-1    linear of order 2
ρ3111111-1-1-1-1111111111111    linear of order 2
ρ41111-1-1-111-111111111-1-1-1-1    linear of order 2
ρ511-1-1-11i-ii-i111111111-1-11    linear of order 4
ρ611-1-11-1ii-i-i11111111-111-1    linear of order 4
ρ711-1-1-11-ii-ii111111111-1-11    linear of order 4
ρ811-1-11-1-i-iii11111111-111-1    linear of order 4
ρ92-2-220000002222-2-2-2-20000    orthogonal lifted from D4
ρ102-22-20000002222-2-2-2-20000    symplectic lifted from Q8, Schur index 2
ρ114400-400000-14-1-14-1-1-10110    orthogonal lifted from C2×F5
ρ1244000-400004-1-1-1-14-1-11001    orthogonal lifted from C2×F5
ρ134400400000-14-1-14-1-1-10-1-10    orthogonal lifted from F5
ρ1444000400004-1-1-1-14-1-1-100-1    orthogonal lifted from F5
ρ154-400000000-14-1-1-41110-√-5√-50    complex lifted from C4⋊F5
ρ164-4000000004-1-1-11-411√-500-√-5    complex lifted from C4⋊F5
ρ174-4000000004-1-1-11-411-√-500√-5    complex lifted from C4⋊F5
ρ184-400000000-14-1-1-41110√-5-√-50    complex lifted from C4⋊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 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)

0-1000000
00-100000
000-10000
11110000
00000-100
000000-10
0000000-1
00001111
,
00001000
0000-1-1-1-1
00000001
00000010
-10000000
11110000
000-10000
00-100000
,
00100000
00010000
-1-1-1-10000
10000000
00000001
0000-1-1-1-1
00001000
00000100
,
10000000
00010000
01000000
-1-1-1-10000
0000-1000
0000000-1
00000-100
00001111

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
 

Export

Character table of Dic5⋊F5 in TeX

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