direct product, metabelian, supersoluble, monomial
Aliases: C5×C32⋊C6, C32⋊C30, He3⋊1C10, C3⋊S3⋊C15, (C3×C15)⋊3S3, (C3×C15)⋊3C6, (C5×He3)⋊4C2, C3.2(S3×C15), C15.6(C3×S3), C32⋊1(C5×S3), (C5×C3⋊S3)⋊C3, SmallGroup(270,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C5×C32⋊C6 |
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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