direct product, metabelian, soluble, monomial, A-group
Aliases: C3×C52⋊C3, C52⋊C32, (C5×C15)⋊C3, SmallGroup(225,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C3×C52⋊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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