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## G = C3×C52⋊S3order 450 = 2·32·52

### Direct product of C3 and C52⋊S3

Aliases: C3×C52⋊S3, (C5×C15)⋊2S3, C52⋊C31C6, C521(C3×S3), (C3×C52⋊C3)⋊3C2, SmallGroup(450,20)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C52⋊C3 — C3×C52⋊S3
 Chief series C1 — C52 — C52⋊C3 — C3×C52⋊C3 — C3×C52⋊S3
 Lower central C52⋊C3 — C3×C52⋊S3
 Upper central C1 — C3

Generators and relations for C3×C52⋊S3
G = < a,b,c,d,e | a3=b5=c5=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc3, be=eb, dcd-1=b-1c3, ece=b-1c-1, ede=d-1 >

15C2
25C3
50C3
3C5
3C5
15C6
25S3
25C32
3D5
15C10
3C15
3C15
25C3×S3
15C30

Smallest permutation representation of C3×C52⋊S3
On 45 points
Generators in S45
(1 34 19)(2 35 20)(3 31 16)(4 32 17)(5 33 18)(6 36 21)(7 37 22)(8 38 23)(9 39 24)(10 40 25)(11 41 26)(12 42 27)(13 43 28)(14 44 29)(15 45 30)
(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)(26 27 28 29 30)(31 32 33 34 35)(36 37 38 39 40)(41 42 43 44 45)
(1 3 5 2 4)(6 10 9 8 7)(16 18 20 17 19)(21 25 24 23 22)(31 33 35 32 34)(36 40 39 38 37)
(1 12 7)(2 15 10)(3 13 8)(4 11 6)(5 14 9)(16 28 23)(17 26 21)(18 29 24)(19 27 22)(20 30 25)(31 43 38)(32 41 36)(33 44 39)(34 42 37)(35 45 40)
(6 11)(7 12)(8 13)(9 14)(10 15)(21 26)(22 27)(23 28)(24 29)(25 30)(36 41)(37 42)(38 43)(39 44)(40 45)

G:=sub<Sym(45)| (1,34,19)(2,35,20)(3,31,16)(4,32,17)(5,33,18)(6,36,21)(7,37,22)(8,38,23)(9,39,24)(10,40,25)(11,41,26)(12,42,27)(13,43,28)(14,44,29)(15,45,30), (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)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45), (1,3,5,2,4)(6,10,9,8,7)(16,18,20,17,19)(21,25,24,23,22)(31,33,35,32,34)(36,40,39,38,37), (1,12,7)(2,15,10)(3,13,8)(4,11,6)(5,14,9)(16,28,23)(17,26,21)(18,29,24)(19,27,22)(20,30,25)(31,43,38)(32,41,36)(33,44,39)(34,42,37)(35,45,40), (6,11)(7,12)(8,13)(9,14)(10,15)(21,26)(22,27)(23,28)(24,29)(25,30)(36,41)(37,42)(38,43)(39,44)(40,45)>;

G:=Group( (1,34,19)(2,35,20)(3,31,16)(4,32,17)(5,33,18)(6,36,21)(7,37,22)(8,38,23)(9,39,24)(10,40,25)(11,41,26)(12,42,27)(13,43,28)(14,44,29)(15,45,30), (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)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45), (1,3,5,2,4)(6,10,9,8,7)(16,18,20,17,19)(21,25,24,23,22)(31,33,35,32,34)(36,40,39,38,37), (1,12,7)(2,15,10)(3,13,8)(4,11,6)(5,14,9)(16,28,23)(17,26,21)(18,29,24)(19,27,22)(20,30,25)(31,43,38)(32,41,36)(33,44,39)(34,42,37)(35,45,40), (6,11)(7,12)(8,13)(9,14)(10,15)(21,26)(22,27)(23,28)(24,29)(25,30)(36,41)(37,42)(38,43)(39,44)(40,45) );

G=PermutationGroup([[(1,34,19),(2,35,20),(3,31,16),(4,32,17),(5,33,18),(6,36,21),(7,37,22),(8,38,23),(9,39,24),(10,40,25),(11,41,26),(12,42,27),(13,43,28),(14,44,29),(15,45,30)], [(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),(26,27,28,29,30),(31,32,33,34,35),(36,37,38,39,40),(41,42,43,44,45)], [(1,3,5,2,4),(6,10,9,8,7),(16,18,20,17,19),(21,25,24,23,22),(31,33,35,32,34),(36,40,39,38,37)], [(1,12,7),(2,15,10),(3,13,8),(4,11,6),(5,14,9),(16,28,23),(17,26,21),(18,29,24),(19,27,22),(20,30,25),(31,43,38),(32,41,36),(33,44,39),(34,42,37),(35,45,40)], [(6,11),(7,12),(8,13),(9,14),(10,15),(21,26),(22,27),(23,28),(24,29),(25,30),(36,41),(37,42),(38,43),(39,44),(40,45)]])

39 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 5A 5B 5C 5D 5E 5F 6A 6B 10A 10B 10C 10D 15A ··· 15H 15I 15J 15K 15L 30A ··· 30H order 1 2 3 3 3 3 3 5 5 5 5 5 5 6 6 10 10 10 10 15 ··· 15 15 15 15 15 30 ··· 30 size 1 15 1 1 50 50 50 3 3 3 3 6 6 15 15 15 15 15 15 3 ··· 3 6 6 6 6 15 ··· 15

39 irreducible representations

 dim 1 1 1 1 2 2 3 3 6 6 type + + + + image C1 C2 C3 C6 S3 C3×S3 C52⋊S3 C3×C52⋊S3 C52⋊S3 C3×C52⋊S3 kernel C3×C52⋊S3 C3×C52⋊C3 C52⋊S3 C52⋊C3 C5×C15 C52 C3 C1 C3 C1 # reps 1 1 2 2 1 2 8 16 2 4

Matrix representation of C3×C52⋊S3 in GL3(𝔽31) generated by

 5 0 0 0 5 0 0 0 5
,
 8 0 0 0 2 0 0 0 2
,
 2 0 0 0 16 0 0 0 1
,
 0 1 0 0 0 1 1 0 0
,
 1 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(31))| [5,0,0,0,5,0,0,0,5],[8,0,0,0,2,0,0,0,2],[2,0,0,0,16,0,0,0,1],[0,0,1,1,0,0,0,1,0],[1,0,0,0,0,1,0,1,0] >;

C3×C52⋊S3 in GAP, Magma, Sage, TeX

C_3\times C_5^2\rtimes S_3
% in TeX

G:=Group("C3xC5^2:S3");
// GroupNames label

G:=SmallGroup(450,20);
// by ID

G=gap.SmallGroup(450,20);
# by ID

G:=PCGroup([5,-2,-3,-3,-5,5,182,973,10804,1439]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^5=c^5=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^3,b*e=e*b,d*c*d^-1=b^-1*c^3,e*c*e=b^-1*c^-1,e*d*e=d^-1>;
// generators/relations

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