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## G = C3×C52⋊C3order 225 = 32·52

### Direct product of C3 and C52⋊C3

Aliases: C3×C52⋊C3, C52⋊C32, (C5×C15)⋊C3, SmallGroup(225,5)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C52⋊C3
G = < a,b,c,d | a3=b5=c5=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3c3, dcd-1=b-1c >

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

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

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

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

C3×C52⋊C3 is a maximal subgroup of   C52⋊(C3⋊S3)  C5⋊D15⋊C3

33 conjugacy classes

 class 1 3A 3B 3C ··· 3H 5A ··· 5H 15A ··· 15P order 1 3 3 3 ··· 3 5 ··· 5 15 ··· 15 size 1 1 1 25 ··· 25 3 ··· 3 3 ··· 3

33 irreducible representations

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

Matrix representation of C3×C52⋊C3 in GL4(𝔽31) generated by

 25 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 4 0 0 0 0 8 0 0 0 0 1
,
 1 0 0 0 0 8 0 0 0 0 2 0 0 0 0 2
,
 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0
G:=sub<GL(4,GF(31))| [25,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,4,0,0,0,0,8,0,0,0,0,1],[1,0,0,0,0,8,0,0,0,0,2,0,0,0,0,2],[1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;

C3×C52⋊C3 in GAP, Magma, Sage, TeX

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

G:=Group("C3xC5^2:C3");
// GroupNames label

G:=SmallGroup(225,5);
// by ID

G=gap.SmallGroup(225,5);
# by ID

G:=PCGroup([4,-3,-3,-5,5,1730,2739]);
// Polycyclic

G:=Group<a,b,c,d|a^3=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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