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G = C5×F5order 100 = 22·52

Direct product of C5 and F5

direct product, metacyclic, supersoluble, monomial, A-group

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

Series: Derived Chief Lower central Upper central

C1C5 — C5×F5
C1C5D5C5×D5 — C5×F5
C5 — C5×F5
C1C5

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

5C2
4C5
5C4
5C10
5C20

Character table of C5×F5

 class 124A4B5A5B5C5D5E5F5G5H5I10A10B10C10D20A20B20C20D20E20F20G20H
 size 1555111144444555555555555
ρ11111111111111111111111111    trivial
ρ211-1-11111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31-1i-i111111111-1-1-1-1-ii-ii-ii-ii    linear of order 4
ρ41-1-ii111111111-1-1-1-1i-ii-ii-ii-i    linear of order 4
ρ511-1-1ζ52ζ53ζ5ζ54ζ5ζ53ζ54ζ521ζ5ζ52ζ53ζ5453535554545252    linear of order 10
ρ61111ζ5ζ54ζ53ζ52ζ53ζ54ζ52ζ51ζ53ζ5ζ54ζ52ζ54ζ54ζ53ζ53ζ52ζ52ζ5ζ5    linear of order 5
ρ71111ζ54ζ5ζ52ζ53ζ52ζ5ζ53ζ541ζ52ζ54ζ5ζ53ζ5ζ5ζ52ζ52ζ53ζ53ζ54ζ54    linear of order 5
ρ811-1-1ζ5ζ54ζ53ζ52ζ53ζ54ζ52ζ51ζ53ζ5ζ54ζ5254545353525255    linear of order 10
ρ91111ζ52ζ53ζ5ζ54ζ5ζ53ζ54ζ521ζ5ζ52ζ53ζ54ζ53ζ53ζ5ζ5ζ54ζ54ζ52ζ52    linear of order 5
ρ101111ζ53ζ52ζ54ζ5ζ54ζ52ζ5ζ531ζ54ζ53ζ52ζ5ζ52ζ52ζ54ζ54ζ5ζ5ζ53ζ53    linear of order 5
ρ1111-1-1ζ53ζ52ζ54ζ5ζ54ζ52ζ5ζ531ζ54ζ53ζ52ζ552525454555353    linear of order 10
ρ1211-1-1ζ54ζ5ζ52ζ53ζ52ζ5ζ53ζ541ζ52ζ54ζ5ζ5355525253535454    linear of order 10
ρ131-1i-iζ5ζ54ζ53ζ52ζ53ζ54ζ52ζ515355452ζ43ζ54ζ4ζ54ζ43ζ53ζ4ζ53ζ43ζ52ζ4ζ52ζ43ζ5ζ4ζ5    linear of order 20
ρ141-1i-iζ52ζ53ζ5ζ54ζ5ζ53ζ54ζ5215525354ζ43ζ53ζ4ζ53ζ43ζ5ζ4ζ5ζ43ζ54ζ4ζ54ζ43ζ52ζ4ζ52    linear of order 20
ρ151-1-iiζ53ζ52ζ54ζ5ζ54ζ52ζ5ζ5315453525ζ4ζ52ζ43ζ52ζ4ζ54ζ43ζ54ζ4ζ5ζ43ζ5ζ4ζ53ζ43ζ53    linear of order 20
ρ161-1-iiζ5ζ54ζ53ζ52ζ53ζ54ζ52ζ515355452ζ4ζ54ζ43ζ54ζ4ζ53ζ43ζ53ζ4ζ52ζ43ζ52ζ4ζ5ζ43ζ5    linear of order 20
ρ171-1-iiζ54ζ5ζ52ζ53ζ52ζ5ζ53ζ5415254553ζ4ζ5ζ43ζ5ζ4ζ52ζ43ζ52ζ4ζ53ζ43ζ53ζ4ζ54ζ43ζ54    linear of order 20
ρ181-1-iiζ52ζ53ζ5ζ54ζ5ζ53ζ54ζ5215525354ζ4ζ53ζ43ζ53ζ4ζ5ζ43ζ5ζ4ζ54ζ43ζ54ζ4ζ52ζ43ζ52    linear of order 20
ρ191-1i-iζ54ζ5ζ52ζ53ζ52ζ5ζ53ζ5415254553ζ43ζ5ζ4ζ5ζ43ζ52ζ4ζ52ζ43ζ53ζ4ζ53ζ43ζ54ζ4ζ54    linear of order 20
ρ201-1i-iζ53ζ52ζ54ζ5ζ54ζ52ζ5ζ5315453525ζ43ζ52ζ4ζ52ζ43ζ54ζ4ζ54ζ43ζ5ζ4ζ5ζ43ζ53ζ4ζ53    linear of order 20
ρ2140004444-1-1-1-1-1000000000000    orthogonal lifted from F5
ρ22400054552535255354-1000000000000    complex faithful
ρ23400053525455452553-1000000000000    complex faithful
ρ24400055453525354525-1000000000000    complex faithful
ρ25400052535545535452-1000000000000    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 9 14 19 25)(2 10 15 20 21)(3 6 11 16 22)(4 7 12 17 23)(5 8 13 18 24)
(6 11 22 16)(7 12 23 17)(8 13 24 18)(9 14 25 19)(10 15 21 20)

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

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

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

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

18000
01800
00180
00018
,
18000
01600
00370
6251010
,
0010
40404015
0100
0001
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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Subgroup lattice of C5×F5 in TeX
Character table of C5×F5 in TeX

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