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## G = C5×C52⋊C3order 375 = 3·53

### Direct product of C5 and C52⋊C3

Aliases: C5×C52⋊C3, C5C3, AΣL1(𝔽125), C53⋊C3, C522C15, SmallGroup(375,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C5×C52⋊C3
 Chief series C1 — C52 — C53 — C5×C52⋊C3
 Lower central C52 — C5×C52⋊C3
 Upper central C1 — C5

Generators and relations for C5×C52⋊C3
G = < a,b,c,d | a5=b5=c5=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3c3, dcd-1=b-1c >

25C3
3C5
3C5
3C5
3C5
3C5
3C5
3C5
3C5
3C5
3C5
25C15
3C52
3C52
3C52
3C52
3C52
3C52
3C52
3C52
3C52
3C52

Permutation representations of C5×C52⋊C3
On 15 points - transitive group 15T25
Generators in S15
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)
(1 3 5 2 4)(6 7 8 9 10)(11 13 15 12 14)
(1 3 5 2 4)(6 9 7 10 8)
(1 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)

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

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

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

G:=TransitiveGroup(15,25);

Polynomial with Galois group C5×C52⋊C3 over ℚ
actionf(x)Disc(f)
15T25x15+5x14-110x13-470x12+3330x11+13297x10-24280x9-114730x8+31690x7+333260x6+13487x5-436445x4-33660x3+263540x2+8415x-57107524·710·414·1672·2232·2812·11354832

55 conjugacy classes

 class 1 3A 3B 5A 5B 5C 5D 5E ··· 5AR 15A ··· 15H order 1 3 3 5 5 5 5 5 ··· 5 15 ··· 15 size 1 25 25 1 1 1 1 3 ··· 3 25 ··· 25

55 irreducible representations

 dim 1 1 1 1 3 3 type + image C1 C3 C5 C15 C52⋊C3 C5×C52⋊C3 kernel C5×C52⋊C3 C53 C52⋊C3 C52 C5 C1 # reps 1 2 4 8 8 32

Matrix representation of C5×C52⋊C3 in GL3(𝔽11) generated by

 3 0 0 0 3 0 0 0 3
,
 4 0 0 0 5 0 0 0 5
,
 9 0 0 0 5 0 0 0 1
,
 0 0 1 1 0 0 0 1 0
G:=sub<GL(3,GF(11))| [3,0,0,0,3,0,0,0,3],[4,0,0,0,5,0,0,0,5],[9,0,0,0,5,0,0,0,1],[0,1,0,0,0,1,1,0,0] >;

C5×C52⋊C3 in GAP, Magma, Sage, TeX

C_5\times C_5^2\rtimes C_3
% in TeX

G:=Group("C5xC5^2:C3");
// GroupNames label

G:=SmallGroup(375,6);
// by ID

G=gap.SmallGroup(375,6);
# by ID

G:=PCGroup([4,-3,-5,-5,5,2882,4563]);
// Polycyclic

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

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