direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C5×C13⋊C3, C65⋊C3, C13⋊C15, SmallGroup(195,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C5×C13⋊C3 |
Generators and relations for C5×C13⋊C3
G = < a,b,c | a5=b13=c3=1, ab=ba, ac=ca, cbc-1=b9 >
Smallest permutation representation of C5×C13⋊C3
►On 65 points
Generators in S65
(1 53 40 27 14)(2 54 41 28 15)(3 55 42 29 16)(4 56 43 30 17)(5 57 44 31 18)(6 58 45 32 19)(7 59 46 33 20)(8 60 47 34 21)(9 61 48 35 22)(10 62 49 36 23)(11 63 50 37 24)(12 64 51 38 25)(13 65 52 39 26)
(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 46 47 48 49 50 51 52)(53 54 55 56 57 58 59 60 61 62 63 64 65)
(2 4 10)(3 7 6)(5 13 11)(8 9 12)(15 17 23)(16 20 19)(18 26 24)(21 22 25)(28 30 36)(29 33 32)(31 39 37)(34 35 38)(41 43 49)(42 46 45)(44 52 50)(47 48 51)(54 56 62)(55 59 58)(57 65 63)(60 61 64)
G:=sub<Sym(65)| (1,53,40,27,14)(2,54,41,28,15)(3,55,42,29,16)(4,56,43,30,17)(5,57,44,31,18)(6,58,45,32,19)(7,59,46,33,20)(8,60,47,34,21)(9,61,48,35,22)(10,62,49,36,23)(11,63,50,37,24)(12,64,51,38,25)(13,65,52,39,26), (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)(40,41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64,65), (2,4,10)(3,7,6)(5,13,11)(8,9,12)(15,17,23)(16,20,19)(18,26,24)(21,22,25)(28,30,36)(29,33,32)(31,39,37)(34,35,38)(41,43,49)(42,46,45)(44,52,50)(47,48,51)(54,56,62)(55,59,58)(57,65,63)(60,61,64)>;
G:=Group( (1,53,40,27,14)(2,54,41,28,15)(3,55,42,29,16)(4,56,43,30,17)(5,57,44,31,18)(6,58,45,32,19)(7,59,46,33,20)(8,60,47,34,21)(9,61,48,35,22)(10,62,49,36,23)(11,63,50,37,24)(12,64,51,38,25)(13,65,52,39,26), (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)(40,41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64,65), (2,4,10)(3,7,6)(5,13,11)(8,9,12)(15,17,23)(16,20,19)(18,26,24)(21,22,25)(28,30,36)(29,33,32)(31,39,37)(34,35,38)(41,43,49)(42,46,45)(44,52,50)(47,48,51)(54,56,62)(55,59,58)(57,65,63)(60,61,64) );
G=PermutationGroup([[(1,53,40,27,14),(2,54,41,28,15),(3,55,42,29,16),(4,56,43,30,17),(5,57,44,31,18),(6,58,45,32,19),(7,59,46,33,20),(8,60,47,34,21),(9,61,48,35,22),(10,62,49,36,23),(11,63,50,37,24),(12,64,51,38,25),(13,65,52,39,26)], [(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),(40,41,42,43,44,45,46,47,48,49,50,51,52),(53,54,55,56,57,58,59,60,61,62,63,64,65)], [(2,4,10),(3,7,6),(5,13,11),(8,9,12),(15,17,23),(16,20,19),(18,26,24),(21,22,25),(28,30,36),(29,33,32),(31,39,37),(34,35,38),(41,43,49),(42,46,45),(44,52,50),(47,48,51),(54,56,62),(55,59,58),(57,65,63),(60,61,64)]])
C5×C13⋊C3 is a maximal subgroup of
D65⋊C3
35 conjugacy classes
| class | 1 | 3A | 3B | 5A | 5B | 5C | 5D | 13A | 13B | 13C | 13D | 15A | ··· | 15H | 65A | ··· | 65P |
| order | 1 | 3 | 3 | 5 | 5 | 5 | 5 | 13 | 13 | 13 | 13 | 15 | ··· | 15 | 65 | ··· | 65 |
| size | 1 | 13 | 13 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 13 | ··· | 13 | 3 | ··· | 3 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | |||||
| image | C1 | C3 | C5 | C15 | C13⋊C3 | C5×C13⋊C3 |
| kernel | C5×C13⋊C3 | C65 | C13⋊C3 | C13 | C5 | C1 |
| # reps | 1 | 2 | 4 | 8 | 4 | 16 |
Matrix representation of C5×C13⋊C3 ►in GL3(𝔽1171) generated by
| 216 | 0 | 0 |
| 0 | 216 | 0 |
| 0 | 0 | 216 |
| 1077 | 852 | 1 |
| 710 | 525 | 1151 |
| 249 | 410 | 832 |
| 1169 | 931 | 460 |
| 838 | 75 | 865 |
| 821 | 790 | 1098 |
G:=sub<GL(3,GF(1171))| [216,0,0,0,216,0,0,0,216],[1077,710,249,852,525,410,1,1151,832],[1169,838,821,931,75,790,460,865,1098] >;
C5×C13⋊C3 in GAP, Magma, Sage, TeX
C_5\times C_{13}\rtimes C_3 % in TeX
G:=Group("C5xC13:C3"); // GroupNames label
G:=SmallGroup(195,1);
// by ID
G=gap.SmallGroup(195,1);
# by ID
G:=PCGroup([3,-3,-5,-13,407]);
// Polycyclic
G:=Group<a,b,c|a^5=b^13=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^9>;
// generators/relations
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