direct product, metacyclic, supersoluble, monomial, Z-group
Aliases: C5×C13⋊C6, C13⋊C30, C65⋊2C6, D13⋊C15, C13⋊C3⋊C10, (C5×D13)⋊C3, (C5×C13⋊C3)⋊2C2, SmallGroup(390,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C5×C13⋊C6 |
Generators and relations for C5×C13⋊C6
G = < a,b,c | a5=b13=c6=1, ab=ba, ac=ca, cbc-1=b10 >
Smallest permutation representation of C5×C13⋊C6
►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 5 4 13 10 11)(3 9 7 12 6 8)(15 18 17 26 23 24)(16 22 20 25 19 21)(28 31 30 39 36 37)(29 35 33 38 32 34)(41 44 43 52 49 50)(42 48 46 51 45 47)(54 57 56 65 62 63)(55 61 59 64 58 60)
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), (2,5,4,13,10,11)(3,9,7,12,6,8)(15,18,17,26,23,24)(16,22,20,25,19,21)(28,31,30,39,36,37)(29,35,33,38,32,34)(41,44,43,52,49,50)(42,48,46,51,45,47)(54,57,56,65,62,63)(55,61,59,64,58,60)>;
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), (2,5,4,13,10,11)(3,9,7,12,6,8)(15,18,17,26,23,24)(16,22,20,25,19,21)(28,31,30,39,36,37)(29,35,33,38,32,34)(41,44,43,52,49,50)(42,48,46,51,45,47)(54,57,56,65,62,63)(55,61,59,64,58,60) );
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)], [(2,5,4,13,10,11),(3,9,7,12,6,8),(15,18,17,26,23,24),(16,22,20,25,19,21),(28,31,30,39,36,37),(29,35,33,38,32,34),(41,44,43,52,49,50),(42,48,46,51,45,47),(54,57,56,65,62,63),(55,61,59,64,58,60)]])
40 conjugacy classes
| class | 1 | 2 | 3A | 3B | 5A | 5B | 5C | 5D | 6A | 6B | 10A | 10B | 10C | 10D | 13A | 13B | 15A | ··· | 15H | 30A | ··· | 30H | 65A | ··· | 65H |
| order | 1 | 2 | 3 | 3 | 5 | 5 | 5 | 5 | 6 | 6 | 10 | 10 | 10 | 10 | 13 | 13 | 15 | ··· | 15 | 30 | ··· | 30 | 65 | ··· | 65 |
| size | 1 | 13 | 13 | 13 | 1 | 1 | 1 | 1 | 13 | 13 | 13 | 13 | 13 | 13 | 6 | 6 | 13 | ··· | 13 | 13 | ··· | 13 | 6 | ··· | 6 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 |
| type | + | + | + | |||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C30 | C13⋊C6 | C5×C13⋊C6 |
| kernel | C5×C13⋊C6 | C5×C13⋊C3 | C5×D13 | C13⋊C6 | C65 | C13⋊C3 | D13 | C13 | C5 | C1 |
| # reps | 1 | 1 | 2 | 4 | 2 | 4 | 8 | 8 | 2 | 8 |
Matrix representation of C5×C13⋊C6 ►in GL6(𝔽1171)
| 70 | 0 | 0 | 0 | 0 | 0 |
| 0 | 70 | 0 | 0 | 0 | 0 |
| 0 | 0 | 70 | 0 | 0 | 0 |
| 0 | 0 | 0 | 70 | 0 | 0 |
| 0 | 0 | 0 | 0 | 70 | 0 |
| 0 | 0 | 0 | 0 | 0 | 70 |
| 741 | 429 | 742 | 430 | 740 | 1170 |
| 742 | 429 | 742 | 430 | 740 | 1170 |
| 741 | 430 | 742 | 430 | 740 | 1170 |
| 741 | 429 | 743 | 430 | 740 | 1170 |
| 741 | 429 | 742 | 431 | 740 | 1170 |
| 741 | 429 | 742 | 430 | 741 | 1170 |
| 1170 | 2 | 430 | 740 | 431 | 2 |
| 431 | 741 | 429 | 742 | 430 | 740 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 430 | 310 | 860 | 743 | 429 | 740 |
| 429 | 743 | 860 | 310 | 430 | 2 |
G:=sub<GL(6,GF(1171))| [70,0,0,0,0,0,0,70,0,0,0,0,0,0,70,0,0,0,0,0,0,70,0,0,0,0,0,0,70,0,0,0,0,0,0,70],[741,742,741,741,741,741,429,429,430,429,429,429,742,742,742,743,742,742,430,430,430,430,431,430,740,740,740,740,740,741,1170,1170,1170,1170,1170,1170],[1170,431,0,1,430,429,2,741,0,0,310,743,430,429,0,0,860,860,740,742,1,0,743,310,431,430,0,0,429,430,2,740,0,0,740,2] >;
C5×C13⋊C6 in GAP, Magma, Sage, TeX
C_5\times C_{13}\rtimes C_6 % in TeX
G:=Group("C5xC13:C6"); // GroupNames label
G:=SmallGroup(390,1);
// by ID
G=gap.SmallGroup(390,1);
# by ID
G:=PCGroup([4,-2,-3,-5,-13,5763,727]);
// Polycyclic
G:=Group<a,b,c|a^5=b^13=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^10>;
// generators/relations
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