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G = C25⋊C4order 100 = 22·52

The semidirect product of C25 and C4 acting faithfully

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C25⋊C4, C5.F5, D25.C2, SmallGroup(100,3)

Series: Derived Chief Lower central Upper central

C1C25 — C25⋊C4
C1C5C25D25 — C25⋊C4
C25 — C25⋊C4
C1

Generators and relations for C25⋊C4
 G = < a,b | a25=b4=1, bab-1=a18 >

25C2
25C4
5D5
5F5

Character table of C25⋊C4

 class 124A4B525A25B25C25D25E
 size 1252525444444
ρ11111111111    trivial
ρ211-1-1111111    linear of order 2
ρ31-1-ii111111    linear of order 4
ρ41-1i-i111111    linear of order 4
ρ540004-1-1-1-1-1    orthogonal lifted from F5
ρ64000-1ζ251625132512259ζ2524251825725ζ25222521254253ζ25192517258256ζ252325142511252    orthogonal faithful
ρ74000-1ζ2524251825725ζ252325142511252ζ25192517258256ζ251625132512259ζ25222521254253    orthogonal faithful
ρ84000-1ζ25192517258256ζ251625132512259ζ252325142511252ζ25222521254253ζ2524251825725    orthogonal faithful
ρ94000-1ζ25222521254253ζ25192517258256ζ2524251825725ζ252325142511252ζ251625132512259    orthogonal faithful
ρ104000-1ζ252325142511252ζ25222521254253ζ251625132512259ζ2524251825725ζ25192517258256    orthogonal faithful

Permutation representations of C25⋊C4
On 25 points - transitive group 25T8
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)
(2 8 25 19)(3 15 24 12)(4 22 23 5)(6 11 21 16)(7 18 20 9)(10 14 17 13)

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

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

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

G:=TransitiveGroup(25,8);

C25⋊C4 is a maximal subgroup of   C75⋊C4  C125⋊C4  C25⋊C20  D25.D5  C25⋊F5  C252F5
C25⋊C4 is a maximal quotient of   C25⋊C8  C75⋊C4  C125⋊C4  D25.D5  C25⋊F5  C252F5

Matrix representation of C25⋊C4 in GL4(𝔽7) generated by

5255
6023
5452
6116
,
0130
0612
2432
2635
G:=sub<GL(4,GF(7))| [5,6,5,6,2,0,4,1,5,2,5,1,5,3,2,6],[0,0,2,2,1,6,4,6,3,1,3,3,0,2,2,5] >;

C25⋊C4 in GAP, Magma, Sage, TeX

C_{25}\rtimes C_4
% in TeX

G:=Group("C25:C4");
// GroupNames label

G:=SmallGroup(100,3);
// by ID

G=gap.SmallGroup(100,3);
# by ID

G:=PCGroup([4,-2,-2,-5,-5,8,338,582,70,643,647]);
// Polycyclic

G:=Group<a,b|a^25=b^4=1,b*a*b^-1=a^18>;
// generators/relations

Export

Subgroup lattice of C25⋊C4 in TeX
Character table of C25⋊C4 in TeX

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