metabelian, soluble, monomial, A-group
Aliases: C52⋊C9, (C5×C15).C3, C3.(C52⋊C3), SmallGroup(225,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C52⋊C9 |
Generators and relations for C52⋊C9
G = < a,b,c | a5=b5=c9=1, cbc-1=ab=ba, cac-1=a3b2 >
Smallest permutation representation of C52⋊C9
►On 45 points
Generators in S45
(2 33 10 41 27)(3 19 42 11 34)(5 36 13 44 21)(6 22 45 14 28)(8 30 16 38 24)(9 25 39 17 31)
(1 40 32 26 18)(2 41 33 27 10)(3 19 42 11 34)(4 43 35 20 12)(5 44 36 21 13)(6 22 45 14 28)(7 37 29 23 15)(8 38 30 24 16)(9 25 39 17 31)
(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)
G:=sub<Sym(45)| (2,33,10,41,27)(3,19,42,11,34)(5,36,13,44,21)(6,22,45,14,28)(8,30,16,38,24)(9,25,39,17,31), (1,40,32,26,18)(2,41,33,27,10)(3,19,42,11,34)(4,43,35,20,12)(5,44,36,21,13)(6,22,45,14,28)(7,37,29,23,15)(8,38,30,24,16)(9,25,39,17,31), (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)>;
G:=Group( (2,33,10,41,27)(3,19,42,11,34)(5,36,13,44,21)(6,22,45,14,28)(8,30,16,38,24)(9,25,39,17,31), (1,40,32,26,18)(2,41,33,27,10)(3,19,42,11,34)(4,43,35,20,12)(5,44,36,21,13)(6,22,45,14,28)(7,37,29,23,15)(8,38,30,24,16)(9,25,39,17,31), (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) );
G=PermutationGroup([[(2,33,10,41,27),(3,19,42,11,34),(5,36,13,44,21),(6,22,45,14,28),(8,30,16,38,24),(9,25,39,17,31)], [(1,40,32,26,18),(2,41,33,27,10),(3,19,42,11,34),(4,43,35,20,12),(5,44,36,21,13),(6,22,45,14,28),(7,37,29,23,15),(8,38,30,24,16),(9,25,39,17,31)], [(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)]])
C52⋊C9 is a maximal subgroup of
C52⋊D9 C52⋊C18
33 conjugacy classes
| class | 1 | 3A | 3B | 5A | ··· | 5H | 9A | ··· | 9F | 15A | ··· | 15P |
| order | 1 | 3 | 3 | 5 | ··· | 5 | 9 | ··· | 9 | 15 | ··· | 15 |
| size | 1 | 1 | 1 | 3 | ··· | 3 | 25 | ··· | 25 | 3 | ··· | 3 |
33 irreducible representations
| dim | 1 | 1 | 1 | 3 | 3 |
| type | + | ||||
| image | C1 | C3 | C9 | C52⋊C3 | C52⋊C9 |
| kernel | C52⋊C9 | C5×C15 | C52 | C3 | C1 |
| # reps | 1 | 2 | 6 | 8 | 16 |
Matrix representation of C52⋊C9 ►in GL4(𝔽181) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 51 | 135 | 0 |
| 0 | 68 | 0 | 59 |
| 1 | 0 | 0 | 0 |
| 0 | 125 | 0 | 0 |
| 0 | 174 | 135 | 0 |
| 0 | 0 | 0 | 125 |
| 39 | 0 | 0 | 0 |
| 0 | 58 | 124 | 0 |
| 0 | 58 | 123 | 1 |
| 0 | 59 | 123 | 0 |
G:=sub<GL(4,GF(181))| [1,0,0,0,0,1,51,68,0,0,135,0,0,0,0,59],[1,0,0,0,0,125,174,0,0,0,135,0,0,0,0,125],[39,0,0,0,0,58,58,59,0,124,123,123,0,0,1,0] >;
C52⋊C9 in GAP, Magma, Sage, TeX
C_5^2\rtimes C_9
% in TeX
G:=Group("C5^2:C9"); // GroupNames label
G:=SmallGroup(225,3);
// by ID
G=gap.SmallGroup(225,3);
# by ID
G:=PCGroup([4,-3,-3,-5,5,12,1730,2739]);
// Polycyclic
G:=Group<a,b,c|a^5=b^5=c^9=1,c*b*c^-1=a*b=b*a,c*a*c^-1=a^3*b^2>;
// generators/relations
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