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G = C52⋊F5order 500 = 22·53

3rd semidirect product of C52 and F5 acting faithfully

non-abelian, supersoluble, monomial

Aliases: He53C4, C523F5, He5⋊C2.2C2, C5.2(C5⋊F5), SmallGroup(500,23)

Series: Derived Chief Lower central Upper central

C1C5He5 — C52⋊F5
C1C5C52He5He5⋊C2 — C52⋊F5
He5 — C52⋊F5
C1

Generators and relations for C52⋊F5
 G = < a,b,c,d | a5=b5=c5=d4=1, ab=ba, cac-1=ab-1, dad-1=a3b-1, bc=cb, dbd-1=b-1, dcd-1=c3 >

25C2
5C5
5C5
5C5
5C5
5C5
5C5
125C4
5D5
5D5
5D5
5D5
5D5
5D5
25C10
25F5
25F5
25F5
25F5
25F5
25F5
25Dic5
5C5×D5
5C5×D5
5C5×D5
5C5×D5
5C5×D5
5C5×D5
5D5.D5
5D5.D5
5D5.D5
5D5.D5
5D5.D5
5D5.D5

Character table of C52⋊F5

 class 124A4B5A5B5C5D5E5F5G5H10A10B
 size 125125125222020202020205050
ρ111111111111111    trivial
ρ211-1-11111111111    linear of order 2
ρ31-1i-i11111111-1-1    linear of order 4
ρ41-1-ii11111111-1-1    linear of order 4
ρ5400044-1-1-14-1-100    orthogonal lifted from F5
ρ6400044-1-1-1-1-1400    orthogonal lifted from F5
ρ7400044-1-14-1-1-100    orthogonal lifted from F5
ρ8400044-1-1-1-14-100    orthogonal lifted from F5
ρ9400044-14-1-1-1-100    orthogonal lifted from F5
ρ104000444-1-1-1-1-100    orthogonal lifted from F5
ρ1110200-5+55/2-5-55/2000000-1-5/2-1+5/2    orthogonal faithful
ρ1210200-5-55/2-5+55/2000000-1+5/2-1-5/2    orthogonal faithful
ρ1310-200-5-55/2-5+55/20000001-5/21+5/2    symplectic faithful, Schur index 2
ρ1410-200-5+55/2-5-55/20000001+5/21-5/2    symplectic faithful, Schur index 2

Permutation representations of C52⋊F5
On 25 points - transitive group 25T33
Generators in S25
(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)
(1 3 2 4 5)(6 8 10 7 9)(11 14 12 15 13)(16 17 18 19 20)(21 25 24 23 22)
(1 24 10 11 16)(2 22 9 12 18)(3 23 7 14 17)(4 21 6 15 19)(5 25 8 13 20)
(1 17 10 14)(2 20 9 13)(3 16 7 11)(4 19 6 15)(5 18 8 12)(22 25)(23 24)

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

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

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

G:=TransitiveGroup(25,33);

Matrix representation of C52⋊F5 in GL10(𝔽41)

344000000000
1000000000
0077000000
003440000000
000040340000
0000770000
0000000100
000000403400
0000000010
0000000001
,
0100000000
403400000000
0001000000
004034000000
0000010000
000040340000
0000000100
000000403400
0000000001
000000004034
,
0000000010
0000000001
1000000000
0100000000
0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
,
1000000000
344000000000
0000001000
000000344000
0010000000
003440000000
0000000010
000000003440
0000100000
000034400000

G:=sub<GL(10,GF(41))| [34,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,7,34,0,0,0,0,0,0,0,0,7,40,0,0,0,0,0,0,0,0,0,0,40,7,0,0,0,0,0,0,0,0,34,7,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[0,40,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,34],[0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[1,34,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,40,0,0,1,34,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,40,0,0] >;

C52⋊F5 in GAP, Magma, Sage, TeX

C_5^2\rtimes F_5
% in TeX

G:=Group("C5^2:F5");
// GroupNames label

G:=SmallGroup(500,23);
// by ID

G=gap.SmallGroup(500,23);
# by ID

G:=PCGroup([5,-2,-2,-5,-5,-5,10,122,127,803,808,613,10004]);
// Polycyclic

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

Export

Subgroup lattice of C52⋊F5 in TeX
Character table of C52⋊F5 in TeX

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