direct product, metabelian, nilpotent (class 3), monomial, 3-elementary
Aliases: C5×C3≀C3, He3⋊1C15, C33⋊1C15, C15.2He3, 3- 1+2⋊1C15, (C5×He3)⋊1C3, C3.2(C5×He3), (C32×C15)⋊1C3, C32.1(C3×C15), (C3×C15).1C32, (C5×3- 1+2)⋊1C3, SmallGroup(405,7)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C3≀C3
G = < a,b,c,d,e | a5=b3=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, be=eb, cd=dc, ce=ec, ede-1=bc-1d >
Smallest permutation representation of C5×C3≀C3
►On 45 points
Generators in S45
(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)
(6 44 15)(7 45 11)(8 41 12)(9 42 13)(10 43 14)(26 31 38)(27 32 39)(28 33 40)(29 34 36)(30 35 37)
(1 23 17)(2 24 18)(3 25 19)(4 21 20)(5 22 16)(6 15 44)(7 11 45)(8 12 41)(9 13 42)(10 14 43)(26 31 38)(27 32 39)(28 33 40)(29 34 36)(30 35 37)
(1 8 31)(2 9 32)(3 10 33)(4 6 34)(5 7 35)(11 37 22)(12 38 23)(13 39 24)(14 40 25)(15 36 21)(16 45 30)(17 41 26)(18 42 27)(19 43 28)(20 44 29)
(1 17 23)(2 18 24)(3 19 25)(4 20 21)(5 16 22)(26 38 31)(27 39 32)(28 40 33)(29 36 34)(30 37 35)
G:=sub<Sym(45)| (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), (6,44,15)(7,45,11)(8,41,12)(9,42,13)(10,43,14)(26,31,38)(27,32,39)(28,33,40)(29,34,36)(30,35,37), (1,23,17)(2,24,18)(3,25,19)(4,21,20)(5,22,16)(6,15,44)(7,11,45)(8,12,41)(9,13,42)(10,14,43)(26,31,38)(27,32,39)(28,33,40)(29,34,36)(30,35,37), (1,8,31)(2,9,32)(3,10,33)(4,6,34)(5,7,35)(11,37,22)(12,38,23)(13,39,24)(14,40,25)(15,36,21)(16,45,30)(17,41,26)(18,42,27)(19,43,28)(20,44,29), (1,17,23)(2,18,24)(3,19,25)(4,20,21)(5,16,22)(26,38,31)(27,39,32)(28,40,33)(29,36,34)(30,37,35)>;
G:=Group( (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), (6,44,15)(7,45,11)(8,41,12)(9,42,13)(10,43,14)(26,31,38)(27,32,39)(28,33,40)(29,34,36)(30,35,37), (1,23,17)(2,24,18)(3,25,19)(4,21,20)(5,22,16)(6,15,44)(7,11,45)(8,12,41)(9,13,42)(10,14,43)(26,31,38)(27,32,39)(28,33,40)(29,34,36)(30,35,37), (1,8,31)(2,9,32)(3,10,33)(4,6,34)(5,7,35)(11,37,22)(12,38,23)(13,39,24)(14,40,25)(15,36,21)(16,45,30)(17,41,26)(18,42,27)(19,43,28)(20,44,29), (1,17,23)(2,18,24)(3,19,25)(4,20,21)(5,16,22)(26,38,31)(27,39,32)(28,40,33)(29,36,34)(30,37,35) );
G=PermutationGroup([[(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)], [(6,44,15),(7,45,11),(8,41,12),(9,42,13),(10,43,14),(26,31,38),(27,32,39),(28,33,40),(29,34,36),(30,35,37)], [(1,23,17),(2,24,18),(3,25,19),(4,21,20),(5,22,16),(6,15,44),(7,11,45),(8,12,41),(9,13,42),(10,14,43),(26,31,38),(27,32,39),(28,33,40),(29,34,36),(30,35,37)], [(1,8,31),(2,9,32),(3,10,33),(4,6,34),(5,7,35),(11,37,22),(12,38,23),(13,39,24),(14,40,25),(15,36,21),(16,45,30),(17,41,26),(18,42,27),(19,43,28),(20,44,29)], [(1,17,23),(2,18,24),(3,19,25),(4,20,21),(5,16,22),(26,38,31),(27,39,32),(28,40,33),(29,36,34),(30,37,35)]])
85 conjugacy classes
| class | 1 | 3A | 3B | 3C | ··· | 3J | 3K | 3L | 5A | 5B | 5C | 5D | 9A | 9B | 9C | 9D | 15A | ··· | 15H | 15I | ··· | 15AN | 15AO | ··· | 15AV | 45A | ··· | 45P |
| order | 1 | 3 | 3 | 3 | ··· | 3 | 3 | 3 | 5 | 5 | 5 | 5 | 9 | 9 | 9 | 9 | 15 | ··· | 15 | 15 | ··· | 15 | 15 | ··· | 15 | 45 | ··· | 45 |
| size | 1 | 1 | 1 | 3 | ··· | 3 | 9 | 9 | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 1 | ··· | 1 | 3 | ··· | 3 | 9 | ··· | 9 | 9 | ··· | 9 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | |||||||||||
| image | C1 | C3 | C3 | C3 | C5 | C15 | C15 | C15 | He3 | C3≀C3 | C5×He3 | C5×C3≀C3 |
| kernel | C5×C3≀C3 | C5×He3 | C5×3- 1+2 | C32×C15 | C3≀C3 | He3 | 3- 1+2 | C33 | C15 | C5 | C3 | C1 |
| # reps | 1 | 2 | 4 | 2 | 4 | 8 | 16 | 8 | 2 | 6 | 8 | 24 |
Matrix representation of C5×C3≀C3 ►in GL3(𝔽181) generated by
| 125 | 0 | 0 |
| 0 | 125 | 0 |
| 0 | 0 | 125 |
| 1 | 0 | 0 |
| 109 | 132 | 0 |
| 31 | 0 | 48 |
| 132 | 0 | 0 |
| 0 | 132 | 0 |
| 0 | 0 | 132 |
| 109 | 131 | 0 |
| 19 | 72 | 1 |
| 162 | 110 | 0 |
| 48 | 0 | 0 |
| 0 | 48 | 0 |
| 150 | 0 | 1 |
G:=sub<GL(3,GF(181))| [125,0,0,0,125,0,0,0,125],[1,109,31,0,132,0,0,0,48],[132,0,0,0,132,0,0,0,132],[109,19,162,131,72,110,0,1,0],[48,0,150,0,48,0,0,0,1] >;
C5×C3≀C3 in GAP, Magma, Sage, TeX
C_5\times C_3\wr C_3
% in TeX
G:=Group("C5xC3wrC3"); // GroupNames label
G:=SmallGroup(405,7);
// by ID
G=gap.SmallGroup(405,7);
# by ID
G:=PCGroup([5,-3,-3,-5,-3,-3,481,3603]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^3=c^3=d^3=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^-1,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c^-1*d>;
// generators/relations
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