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## G = C5×C32⋊C6order 270 = 2·33·5

### Direct product of C5 and C32⋊C6

Aliases: C5×C32⋊C6, C32⋊C30, He31C10, C3⋊S3⋊C15, (C3×C15)⋊3S3, (C3×C15)⋊3C6, (C5×He3)⋊4C2, C3.2(S3×C15), C15.6(C3×S3), C321(C5×S3), (C5×C3⋊S3)⋊C3, SmallGroup(270,10)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C5×C32⋊C6
 Chief series C1 — C3 — C32 — C3×C15 — C5×He3 — C5×C32⋊C6
 Lower central C32 — C5×C32⋊C6
 Upper central C1 — C5

Generators and relations for C5×C32⋊C6
G = < a,b,c,d | a5=b3=c3=d6=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c-1, dcd-1=c-1 >

Smallest permutation representation of C5×C32⋊C6
On 45 points
Generators in S45
(1 5 2 4 3)(6 13 8 15 11)(7 12 9 14 10)(16 39 27 31 41)(17 34 22 32 42)(18 35 23 33 43)(19 36 24 28 44)(20 37 25 29 45)(21 38 26 30 40)
(1 43 40)(2 35 38)(3 33 30)(4 23 26)(5 18 21)(6 29 28)(7 31 32)(8 20 19)(9 16 17)(10 27 22)(11 25 24)(12 41 42)(13 45 44)(14 39 34)(15 37 36)
(1 13 12)(2 15 14)(3 6 7)(4 11 10)(5 8 9)(16 18 20)(17 21 19)(22 26 24)(23 25 27)(28 32 30)(29 31 33)(34 38 36)(35 37 39)(40 44 42)(41 43 45)
(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)

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

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

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

50 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 5A 5B 5C 5D 6A 6B 10A 10B 10C 10D 15A 15B 15C 15D 15E ··· 15L 15M ··· 15X 30A ··· 30H order 1 2 3 3 3 3 3 3 5 5 5 5 6 6 10 10 10 10 15 15 15 15 15 ··· 15 15 ··· 15 30 ··· 30 size 1 9 2 3 3 6 6 6 1 1 1 1 9 9 9 9 9 9 2 2 2 2 3 ··· 3 6 ··· 6 9 ··· 9

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 6 6 type + + + + image C1 C2 C3 C5 C6 C10 C15 C30 S3 C3×S3 C5×S3 S3×C15 C32⋊C6 C5×C32⋊C6 kernel C5×C32⋊C6 C5×He3 C5×C3⋊S3 C32⋊C6 C3×C15 He3 C3⋊S3 C32 C3×C15 C15 C32 C3 C5 C1 # reps 1 1 2 4 2 4 8 8 1 2 4 8 1 4

Matrix representation of C5×C32⋊C6 in GL6(𝔽31)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0
,
 30 30 0 0 0 0 1 0 0 0 0 0 0 0 30 30 0 0 0 0 1 0 0 0 0 0 0 0 30 30 0 0 0 0 1 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 30 30 0 0 0 0 0 1 0 0 1 0 0 0 0 0 30 30 0 0

G:=sub<GL(6,GF(31))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[30,1,0,0,0,0,30,0,0,0,0,0,0,0,30,1,0,0,0,0,30,0,0,0,0,0,0,0,30,1,0,0,0,0,30,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,30,0,0,0,0,0,30,0,0,30,0,0,0,0,0,30,1,0,0] >;

C5×C32⋊C6 in GAP, Magma, Sage, TeX

C_5\times C_3^2\rtimes C_6
% in TeX

G:=Group("C5xC3^2:C6");
// GroupNames label

G:=SmallGroup(270,10);
// by ID

G=gap.SmallGroup(270,10);
# by ID

G:=PCGroup([5,-2,-3,-5,-3,-3,1203,1208,4504]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^3=c^3=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c^-1,d*c*d^-1=c^-1>;
// generators/relations

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