direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C13×C7⋊C3, C7⋊C39, C91⋊1C3, SmallGroup(273,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C13×C7⋊C3 |
Generators and relations for C13×C7⋊C3
G = < a,b,c | a13=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >
Smallest permutation representation of C13×C7⋊C3
►On 91 points
Generators in S91
(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)(66 67 68 69 70 71 72 73 74 75 76 77 78)(79 80 81 82 83 84 85 86 87 88 89 90 91)
(1 86 32 64 68 22 51)(2 87 33 65 69 23 52)(3 88 34 53 70 24 40)(4 89 35 54 71 25 41)(5 90 36 55 72 26 42)(6 91 37 56 73 14 43)(7 79 38 57 74 15 44)(8 80 39 58 75 16 45)(9 81 27 59 76 17 46)(10 82 28 60 77 18 47)(11 83 29 61 78 19 48)(12 84 30 62 66 20 49)(13 85 31 63 67 21 50)
(14 56 43)(15 57 44)(16 58 45)(17 59 46)(18 60 47)(19 61 48)(20 62 49)(21 63 50)(22 64 51)(23 65 52)(24 53 40)(25 54 41)(26 55 42)(27 76 81)(28 77 82)(29 78 83)(30 66 84)(31 67 85)(32 68 86)(33 69 87)(34 70 88)(35 71 89)(36 72 90)(37 73 91)(38 74 79)(39 75 80)
G:=sub<Sym(91)| (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)(66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91), (1,86,32,64,68,22,51)(2,87,33,65,69,23,52)(3,88,34,53,70,24,40)(4,89,35,54,71,25,41)(5,90,36,55,72,26,42)(6,91,37,56,73,14,43)(7,79,38,57,74,15,44)(8,80,39,58,75,16,45)(9,81,27,59,76,17,46)(10,82,28,60,77,18,47)(11,83,29,61,78,19,48)(12,84,30,62,66,20,49)(13,85,31,63,67,21,50), (14,56,43)(15,57,44)(16,58,45)(17,59,46)(18,60,47)(19,61,48)(20,62,49)(21,63,50)(22,64,51)(23,65,52)(24,53,40)(25,54,41)(26,55,42)(27,76,81)(28,77,82)(29,78,83)(30,66,84)(31,67,85)(32,68,86)(33,69,87)(34,70,88)(35,71,89)(36,72,90)(37,73,91)(38,74,79)(39,75,80)>;
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)(40,41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64,65)(66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91), (1,86,32,64,68,22,51)(2,87,33,65,69,23,52)(3,88,34,53,70,24,40)(4,89,35,54,71,25,41)(5,90,36,55,72,26,42)(6,91,37,56,73,14,43)(7,79,38,57,74,15,44)(8,80,39,58,75,16,45)(9,81,27,59,76,17,46)(10,82,28,60,77,18,47)(11,83,29,61,78,19,48)(12,84,30,62,66,20,49)(13,85,31,63,67,21,50), (14,56,43)(15,57,44)(16,58,45)(17,59,46)(18,60,47)(19,61,48)(20,62,49)(21,63,50)(22,64,51)(23,65,52)(24,53,40)(25,54,41)(26,55,42)(27,76,81)(28,77,82)(29,78,83)(30,66,84)(31,67,85)(32,68,86)(33,69,87)(34,70,88)(35,71,89)(36,72,90)(37,73,91)(38,74,79)(39,75,80) );
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),(40,41,42,43,44,45,46,47,48,49,50,51,52),(53,54,55,56,57,58,59,60,61,62,63,64,65),(66,67,68,69,70,71,72,73,74,75,76,77,78),(79,80,81,82,83,84,85,86,87,88,89,90,91)], [(1,86,32,64,68,22,51),(2,87,33,65,69,23,52),(3,88,34,53,70,24,40),(4,89,35,54,71,25,41),(5,90,36,55,72,26,42),(6,91,37,56,73,14,43),(7,79,38,57,74,15,44),(8,80,39,58,75,16,45),(9,81,27,59,76,17,46),(10,82,28,60,77,18,47),(11,83,29,61,78,19,48),(12,84,30,62,66,20,49),(13,85,31,63,67,21,50)], [(14,56,43),(15,57,44),(16,58,45),(17,59,46),(18,60,47),(19,61,48),(20,62,49),(21,63,50),(22,64,51),(23,65,52),(24,53,40),(25,54,41),(26,55,42),(27,76,81),(28,77,82),(29,78,83),(30,66,84),(31,67,85),(32,68,86),(33,69,87),(34,70,88),(35,71,89),(36,72,90),(37,73,91),(38,74,79),(39,75,80)]])
65 conjugacy classes
| class | 1 | 3A | 3B | 7A | 7B | 13A | ··· | 13L | 39A | ··· | 39X | 91A | ··· | 91X |
| order | 1 | 3 | 3 | 7 | 7 | 13 | ··· | 13 | 39 | ··· | 39 | 91 | ··· | 91 |
| size | 1 | 7 | 7 | 3 | 3 | 1 | ··· | 1 | 7 | ··· | 7 | 3 | ··· | 3 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | |||||
| image | C1 | C3 | C13 | C39 | C7⋊C3 | C13×C7⋊C3 |
| kernel | C13×C7⋊C3 | C91 | C7⋊C3 | C7 | C13 | C1 |
| # reps | 1 | 2 | 12 | 24 | 2 | 24 |
Matrix representation of C13×C7⋊C3 ►in GL3(𝔽547) generated by
| 475 | 0 | 0 |
| 0 | 475 | 0 |
| 0 | 0 | 475 |
| 459 | 460 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 87 | 546 | 546 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(547))| [475,0,0,0,475,0,0,0,475],[459,1,0,460,0,1,1,0,0],[1,87,0,0,546,1,0,546,0] >;
C13×C7⋊C3 in GAP, Magma, Sage, TeX
C_{13}\times C_7\rtimes C_3 % in TeX
G:=Group("C13xC7:C3"); // GroupNames label
G:=SmallGroup(273,1);
// by ID
G=gap.SmallGroup(273,1);
# by ID
G:=PCGroup([3,-3,-13,-7,704]);
// Polycyclic
G:=Group<a,b,c|a^13=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
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