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## G = S3×C52⋊C3order 450 = 2·32·52

### Direct product of S3 and C52⋊C3

Aliases: S3×C52⋊C3, (C5×C15)⋊3C6, (S3×C52)⋊C3, C523(C3×S3), C3⋊(C2×C52⋊C3), (C3×C52⋊C3)⋊5C2, SmallGroup(450,23)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C15 — S3×C52⋊C3
 Chief series C1 — C52 — C5×C15 — C3×C52⋊C3 — S3×C52⋊C3
 Lower central C5×C15 — S3×C52⋊C3
 Upper central C1

Generators and relations for S3×C52⋊C3
G = < a,b,c,d,e | a3=b2=c5=d5=e3=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c3d3, ede-1=c-1d >

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

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

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

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

33 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 5A ··· 5H 6A 6B 10A ··· 10H 15A ··· 15H order 1 2 3 3 3 3 3 5 ··· 5 6 6 10 ··· 10 15 ··· 15 size 1 3 2 25 25 50 50 3 ··· 3 75 75 9 ··· 9 6 ··· 6

33 irreducible representations

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

Matrix representation of S3×C52⋊C3 in GL5(𝔽31)

 0 30 0 0 0 1 30 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 2 19 0 0 0 0 8 0 0 0 0 0 2
,
 1 0 0 0 0 0 1 0 0 0 0 0 2 3 2 0 0 0 16 0 0 0 0 0 1
,
 25 0 0 0 0 0 25 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 15 30 30

G:=sub<GL(5,GF(31))| [0,1,0,0,0,30,30,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,19,8,0,0,0,0,0,2],[1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,3,16,0,0,0,2,0,1],[25,0,0,0,0,0,25,0,0,0,0,0,1,0,15,0,0,0,0,30,0,0,0,1,30] >;

S3×C52⋊C3 in GAP, Magma, Sage, TeX

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

G:=Group("S3xC5^2:C3");
// GroupNames label

G:=SmallGroup(450,23);
// by ID

G=gap.SmallGroup(450,23);
# by ID

G:=PCGroup([5,-2,-3,-3,-5,5,182,2348,1359]);
// Polycyclic

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

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