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## G = C5×F5order 100 = 22·52

### Direct product of C5 and F5

Aliases: C5×F5, C5⋊C20, D5.C10, C521C4, (C5×D5).1C2, SmallGroup(100,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C5×F5
 Chief series C1 — C5 — D5 — C5×D5 — C5×F5
 Lower central C5 — C5×F5
 Upper central C1 — C5

Generators and relations for C5×F5
G = < a,b,c | a5=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >

Character table of C5×F5

 class 1 2 4A 4B 5A 5B 5C 5D 5E 5F 5G 5H 5I 10A 10B 10C 10D 20A 20B 20C 20D 20E 20F 20G 20H size 1 5 5 5 1 1 1 1 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 ρ1 1 1 1 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 -1 -1 -1 linear of order 2 ρ3 1 -1 i -i 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -i i -i i -i i -i i linear of order 4 ρ4 1 -1 -i i 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 i -i i -i i -i i -i linear of order 4 ρ5 1 1 -1 -1 ζ52 ζ53 ζ5 ζ54 ζ5 ζ53 ζ54 ζ52 1 ζ5 ζ52 ζ53 ζ54 -ζ53 -ζ53 -ζ5 -ζ5 -ζ54 -ζ54 -ζ52 -ζ52 linear of order 10 ρ6 1 1 1 1 ζ5 ζ54 ζ53 ζ52 ζ53 ζ54 ζ52 ζ5 1 ζ53 ζ5 ζ54 ζ52 ζ54 ζ54 ζ53 ζ53 ζ52 ζ52 ζ5 ζ5 linear of order 5 ρ7 1 1 1 1 ζ54 ζ5 ζ52 ζ53 ζ52 ζ5 ζ53 ζ54 1 ζ52 ζ54 ζ5 ζ53 ζ5 ζ5 ζ52 ζ52 ζ53 ζ53 ζ54 ζ54 linear of order 5 ρ8 1 1 -1 -1 ζ5 ζ54 ζ53 ζ52 ζ53 ζ54 ζ52 ζ5 1 ζ53 ζ5 ζ54 ζ52 -ζ54 -ζ54 -ζ53 -ζ53 -ζ52 -ζ52 -ζ5 -ζ5 linear of order 10 ρ9 1 1 1 1 ζ52 ζ53 ζ5 ζ54 ζ5 ζ53 ζ54 ζ52 1 ζ5 ζ52 ζ53 ζ54 ζ53 ζ53 ζ5 ζ5 ζ54 ζ54 ζ52 ζ52 linear of order 5 ρ10 1 1 1 1 ζ53 ζ52 ζ54 ζ5 ζ54 ζ52 ζ5 ζ53 1 ζ54 ζ53 ζ52 ζ5 ζ52 ζ52 ζ54 ζ54 ζ5 ζ5 ζ53 ζ53 linear of order 5 ρ11 1 1 -1 -1 ζ53 ζ52 ζ54 ζ5 ζ54 ζ52 ζ5 ζ53 1 ζ54 ζ53 ζ52 ζ5 -ζ52 -ζ52 -ζ54 -ζ54 -ζ5 -ζ5 -ζ53 -ζ53 linear of order 10 ρ12 1 1 -1 -1 ζ54 ζ5 ζ52 ζ53 ζ52 ζ5 ζ53 ζ54 1 ζ52 ζ54 ζ5 ζ53 -ζ5 -ζ5 -ζ52 -ζ52 -ζ53 -ζ53 -ζ54 -ζ54 linear of order 10 ρ13 1 -1 i -i ζ5 ζ54 ζ53 ζ52 ζ53 ζ54 ζ52 ζ5 1 -ζ53 -ζ5 -ζ54 -ζ52 ζ43ζ54 ζ4ζ54 ζ43ζ53 ζ4ζ53 ζ43ζ52 ζ4ζ52 ζ43ζ5 ζ4ζ5 linear of order 20 ρ14 1 -1 i -i ζ52 ζ53 ζ5 ζ54 ζ5 ζ53 ζ54 ζ52 1 -ζ5 -ζ52 -ζ53 -ζ54 ζ43ζ53 ζ4ζ53 ζ43ζ5 ζ4ζ5 ζ43ζ54 ζ4ζ54 ζ43ζ52 ζ4ζ52 linear of order 20 ρ15 1 -1 -i i ζ53 ζ52 ζ54 ζ5 ζ54 ζ52 ζ5 ζ53 1 -ζ54 -ζ53 -ζ52 -ζ5 ζ4ζ52 ζ43ζ52 ζ4ζ54 ζ43ζ54 ζ4ζ5 ζ43ζ5 ζ4ζ53 ζ43ζ53 linear of order 20 ρ16 1 -1 -i i ζ5 ζ54 ζ53 ζ52 ζ53 ζ54 ζ52 ζ5 1 -ζ53 -ζ5 -ζ54 -ζ52 ζ4ζ54 ζ43ζ54 ζ4ζ53 ζ43ζ53 ζ4ζ52 ζ43ζ52 ζ4ζ5 ζ43ζ5 linear of order 20 ρ17 1 -1 -i i ζ54 ζ5 ζ52 ζ53 ζ52 ζ5 ζ53 ζ54 1 -ζ52 -ζ54 -ζ5 -ζ53 ζ4ζ5 ζ43ζ5 ζ4ζ52 ζ43ζ52 ζ4ζ53 ζ43ζ53 ζ4ζ54 ζ43ζ54 linear of order 20 ρ18 1 -1 -i i ζ52 ζ53 ζ5 ζ54 ζ5 ζ53 ζ54 ζ52 1 -ζ5 -ζ52 -ζ53 -ζ54 ζ4ζ53 ζ43ζ53 ζ4ζ5 ζ43ζ5 ζ4ζ54 ζ43ζ54 ζ4ζ52 ζ43ζ52 linear of order 20 ρ19 1 -1 i -i ζ54 ζ5 ζ52 ζ53 ζ52 ζ5 ζ53 ζ54 1 -ζ52 -ζ54 -ζ5 -ζ53 ζ43ζ5 ζ4ζ5 ζ43ζ52 ζ4ζ52 ζ43ζ53 ζ4ζ53 ζ43ζ54 ζ4ζ54 linear of order 20 ρ20 1 -1 i -i ζ53 ζ52 ζ54 ζ5 ζ54 ζ52 ζ5 ζ53 1 -ζ54 -ζ53 -ζ52 -ζ5 ζ43ζ52 ζ4ζ52 ζ43ζ54 ζ4ζ54 ζ43ζ5 ζ4ζ5 ζ43ζ53 ζ4ζ53 linear of order 20 ρ21 4 0 0 0 4 4 4 4 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from F5 ρ22 4 0 0 0 4ζ54 4ζ5 4ζ52 4ζ53 -ζ52 -ζ5 -ζ53 -ζ54 -1 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ23 4 0 0 0 4ζ53 4ζ52 4ζ54 4ζ5 -ζ54 -ζ52 -ζ5 -ζ53 -1 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ24 4 0 0 0 4ζ5 4ζ54 4ζ53 4ζ52 -ζ53 -ζ54 -ζ52 -ζ5 -1 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ25 4 0 0 0 4ζ52 4ζ53 4ζ5 4ζ54 -ζ5 -ζ53 -ζ54 -ζ52 -1 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C5×F5
On 20 points - transitive group 20T29
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 2 3 4 5)(6 10 9 8 7)(11 13 15 12 14)(16 19 17 20 18)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)

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

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

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

G:=TransitiveGroup(20,29);

On 25 points - transitive group 25T7
Generators in S25
(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)
(1 19 12 9 25)(2 20 13 10 21)(3 16 14 6 22)(4 17 15 7 23)(5 18 11 8 24)
(6 16 14 22)(7 17 15 23)(8 18 11 24)(9 19 12 25)(10 20 13 21)

G:=sub<Sym(25)| (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), (1,19,12,9,25)(2,20,13,10,21)(3,16,14,6,22)(4,17,15,7,23)(5,18,11,8,24), (6,16,14,22)(7,17,15,23)(8,18,11,24)(9,19,12,25)(10,20,13,21)>;

G:=Group( (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), (1,19,12,9,25)(2,20,13,10,21)(3,16,14,6,22)(4,17,15,7,23)(5,18,11,8,24), (6,16,14,22)(7,17,15,23)(8,18,11,24)(9,19,12,25)(10,20,13,21) );

G=PermutationGroup([[(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)], [(1,19,12,9,25),(2,20,13,10,21),(3,16,14,6,22),(4,17,15,7,23),(5,18,11,8,24)], [(6,16,14,22),(7,17,15,23),(8,18,11,24),(9,19,12,25),(10,20,13,21)]])

G:=TransitiveGroup(25,7);

C5×F5 is a maximal subgroup of   C52⋊C20  C25⋊C20  He54C4
C5×F5 is a maximal quotient of   C52⋊C20  C25⋊C20

Matrix representation of C5×F5 in GL4(𝔽41) generated by

 18 0 0 0 0 18 0 0 0 0 18 0 0 0 0 18
,
 18 0 0 0 0 16 0 0 0 0 37 0 6 25 10 10
,
 0 0 1 0 40 40 40 15 0 1 0 0 0 0 0 1
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[18,0,0,6,0,16,0,25,0,0,37,10,0,0,0,10],[0,40,0,0,0,40,1,0,1,40,0,0,0,15,0,1] >;

C5×F5 in GAP, Magma, Sage, TeX

C_5\times F_5
% in TeX

G:=Group("C5xF5");
// GroupNames label

G:=SmallGroup(100,9);
// by ID

G=gap.SmallGroup(100,9);
# by ID

G:=PCGroup([4,-2,-5,-2,-5,40,643,139]);
// Polycyclic

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

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