direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C32xD5, C15:2C6, C5:(C3xC6), (C3xC15):3C2, SmallGroup(90,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C32xD5 |
Generators and relations for C32xD5
G = < a,b,c,d | a3=b3=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Quotients: C1, C2, C3, C6, C32, D5, C3xC6, C3xD5, C32xD5
Smallest permutation representation of C32xD5
►On 45 points
Generators in S45
(1 44 24)(2 45 25)(3 41 21)(4 42 22)(5 43 23)(6 31 26)(7 32 27)(8 33 28)(9 34 29)(10 35 30)(11 36 16)(12 37 17)(13 38 18)(14 39 19)(15 40 20)
(1 14 9)(2 15 10)(3 11 6)(4 12 7)(5 13 8)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)
(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)
(1 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)(42 45)(43 44)
G:=sub<Sym(45)| (1,44,24)(2,45,25)(3,41,21)(4,42,22)(5,43,23)(6,31,26)(7,32,27)(8,33,28)(9,34,29)(10,35,30)(11,36,16)(12,37,17)(13,38,18)(14,39,19)(15,40,20), (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40), (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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39)(42,45)(43,44)>;
G:=Group( (1,44,24)(2,45,25)(3,41,21)(4,42,22)(5,43,23)(6,31,26)(7,32,27)(8,33,28)(9,34,29)(10,35,30)(11,36,16)(12,37,17)(13,38,18)(14,39,19)(15,40,20), (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40), (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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39)(42,45)(43,44) );
G=PermutationGroup([[(1,44,24),(2,45,25),(3,41,21),(4,42,22),(5,43,23),(6,31,26),(7,32,27),(8,33,28),(9,34,29),(10,35,30),(11,36,16),(12,37,17),(13,38,18),(14,39,19),(15,40,20)], [(1,14,9),(2,15,10),(3,11,6),(4,12,7),(5,13,8),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25),(31,41,36),(32,42,37),(33,43,38),(34,44,39),(35,45,40)], [(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)], [(1,5),(2,4),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19),(22,25),(23,24),(27,30),(28,29),(32,35),(33,34),(37,40),(38,39),(42,45),(43,44)]])
C32xD5 is a maximal subgroup of
C32:3F5
36 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 5A | 5B | 6A | ··· | 6H | 15A | ··· | 15P |
| order | 1 | 2 | 3 | ··· | 3 | 5 | 5 | 6 | ··· | 6 | 15 | ··· | 15 |
| size | 1 | 5 | 1 | ··· | 1 | 2 | 2 | 5 | ··· | 5 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D5 | C3xD5 |
| kernel | C32xD5 | C3xC15 | C3xD5 | C15 | C32 | C3 |
| # reps | 1 | 1 | 8 | 8 | 2 | 16 |
Matrix representation of C32xD5 ►in GL3(F31) generated by
| 25 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 5 |
| 1 | 0 | 0 |
| 0 | 30 | 1 |
| 0 | 17 | 13 |
| 30 | 0 | 0 |
| 0 | 30 | 0 |
| 0 | 17 | 1 |
G:=sub<GL(3,GF(31))| [25,0,0,0,1,0,0,0,1],[1,0,0,0,5,0,0,0,5],[1,0,0,0,30,17,0,1,13],[30,0,0,0,30,17,0,0,1] >;
C32xD5 in GAP, Magma, Sage, TeX
C_3^2\times D_5
% in TeX
G:=Group("C3^2xD5"); // GroupNames label
G:=SmallGroup(90,5);
// by ID
G=gap.SmallGroup(90,5);
# by ID
G:=PCGroup([4,-2,-3,-3,-5,1155]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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