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G = C52⋊C3order 75 = 3·52

The semidirect product of C52 and C3 acting faithfully

metabelian, soluble, monomial, A-group

Aliases: C52⋊C3, SmallGroup(75,2)

Series: Derived Chief Lower central Upper central

C1C52 — C52⋊C3
C1C52 — C52⋊C3
C52 — C52⋊C3
C1

Generators and relations for C52⋊C3
 G = < a,b,c | a5=b5=c3=1, ab=ba, cac-1=a3b3, cbc-1=a-1b >

25C3
3C5
3C5

Character table of C52⋊C3

 class 13A3B5A5B5C5D5E5F5G5H
 size 1252533333333
ρ111111111111    trivial
ρ21ζ3ζ3211111111    linear of order 3
ρ31ζ32ζ311111111    linear of order 3
ρ43001+5/21-5/25452ζ54+2ζ53ζ53+2ζ55251-5/21+5/2    complex faithful
ρ5300ζ53+2ζ5ζ54+2ζ531-5/21+5/21-5/21+5/25255452    complex faithful
ρ6300525ζ53+2ζ51+5/21-5/21+5/21-5/25452ζ54+2ζ53    complex faithful
ρ73001+5/21-5/2ζ53+2ζ55255452ζ54+2ζ531-5/21+5/2    complex faithful
ρ83001-5/21+5/2ζ54+2ζ53ζ53+2ζ552554521+5/21-5/2    complex faithful
ρ930054525251-5/21+5/21-5/21+5/2ζ54+2ζ53ζ53+2ζ5    complex faithful
ρ10300ζ54+2ζ5354521+5/21-5/21+5/21-5/2ζ53+2ζ5525    complex faithful
ρ113001-5/21+5/25255452ζ54+2ζ53ζ53+2ζ51+5/21-5/2    complex faithful

Permutation representations of C52⋊C3
On 15 points - transitive group 15T9
Generators in S15
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)
(6 7 8 9 10)(11 14 12 15 13)
(1 13 9)(2 15 10)(3 12 6)(4 14 7)(5 11 8)

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

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

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

G:=TransitiveGroup(15,9);

On 25 points: primitive - transitive group 25T6
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 25 14 9 19)(2 21 15 10 20)(3 22 11 6 16)(4 23 12 7 17)(5 24 13 8 18)
(2 15 7)(3 16 21)(4 23 18)(5 8 11)(6 9 17)(10 24 25)(12 14 22)(13 20 19)

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

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

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

G:=TransitiveGroup(25,6);

C52⋊C3 is a maximal subgroup of   C52⋊S3  C52⋊C6  C52⋊A4
C52⋊C3 is a maximal quotient of   C52⋊C9  C52⋊A4  He5⋊C3

Polynomial with Galois group C52⋊C3 over ℚ
actionf(x)Disc(f)
15T9x15-470x13-305x12+71840x11+85357x10-4292700x9-3714805x8+119761820x7+25284495x6-1542190154x5+717324725x4+7178878600x3-5452953875x2-7998223215x+4461221029224·36·524·750·4118·2932·15672·29272·16079412

Matrix representation of C52⋊C3 in GL3(𝔽11) generated by

100
090
005
,
400
040
009
,
001
100
010
G:=sub<GL(3,GF(11))| [1,0,0,0,9,0,0,0,5],[4,0,0,0,4,0,0,0,9],[0,1,0,0,0,1,1,0,0] >;

C52⋊C3 in GAP, Magma, Sage, TeX

C_5^2\rtimes C_3
% in TeX

G:=Group("C5^2:C3");
// GroupNames label

G:=SmallGroup(75,2);
// by ID

G=gap.SmallGroup(75,2);
# by ID

G:=PCGroup([3,-3,-5,5,199,434]);
// Polycyclic

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

Export

Subgroup lattice of C52⋊C3 in TeX
Character table of C52⋊C3 in TeX

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