direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C13×D7, C7⋊C26, C91⋊3C2, SmallGroup(182,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C13×D7 |
Generators and relations for C13×D7
G = < a,b,c | a13=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C13×D7
►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 88 49 22 69 38 54)(2 89 50 23 70 39 55)(3 90 51 24 71 27 56)(4 91 52 25 72 28 57)(5 79 40 26 73 29 58)(6 80 41 14 74 30 59)(7 81 42 15 75 31 60)(8 82 43 16 76 32 61)(9 83 44 17 77 33 62)(10 84 45 18 78 34 63)(11 85 46 19 66 35 64)(12 86 47 20 67 36 65)(13 87 48 21 68 37 53)
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 61)(9 62)(10 63)(11 64)(12 65)(13 53)(27 90)(28 91)(29 79)(30 80)(31 81)(32 82)(33 83)(34 84)(35 85)(36 86)(37 87)(38 88)(39 89)(40 73)(41 74)(42 75)(43 76)(44 77)(45 78)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)
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), (1,88,49,22,69,38,54)(2,89,50,23,70,39,55)(3,90,51,24,71,27,56)(4,91,52,25,72,28,57)(5,79,40,26,73,29,58)(6,80,41,14,74,30,59)(7,81,42,15,75,31,60)(8,82,43,16,76,32,61)(9,83,44,17,77,33,62)(10,84,45,18,78,34,63)(11,85,46,19,66,35,64)(12,86,47,20,67,36,65)(13,87,48,21,68,37,53), (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,65)(13,53)(27,90)(28,91)(29,79)(30,80)(31,81)(32,82)(33,83)(34,84)(35,85)(36,86)(37,87)(38,88)(39,89)(40,73)(41,74)(42,75)(43,76)(44,77)(45,78)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)>;
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), (1,88,49,22,69,38,54)(2,89,50,23,70,39,55)(3,90,51,24,71,27,56)(4,91,52,25,72,28,57)(5,79,40,26,73,29,58)(6,80,41,14,74,30,59)(7,81,42,15,75,31,60)(8,82,43,16,76,32,61)(9,83,44,17,77,33,62)(10,84,45,18,78,34,63)(11,85,46,19,66,35,64)(12,86,47,20,67,36,65)(13,87,48,21,68,37,53), (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,65)(13,53)(27,90)(28,91)(29,79)(30,80)(31,81)(32,82)(33,83)(34,84)(35,85)(36,86)(37,87)(38,88)(39,89)(40,73)(41,74)(42,75)(43,76)(44,77)(45,78)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72) );
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)], [(1,88,49,22,69,38,54),(2,89,50,23,70,39,55),(3,90,51,24,71,27,56),(4,91,52,25,72,28,57),(5,79,40,26,73,29,58),(6,80,41,14,74,30,59),(7,81,42,15,75,31,60),(8,82,43,16,76,32,61),(9,83,44,17,77,33,62),(10,84,45,18,78,34,63),(11,85,46,19,66,35,64),(12,86,47,20,67,36,65),(13,87,48,21,68,37,53)], [(1,54),(2,55),(3,56),(4,57),(5,58),(6,59),(7,60),(8,61),(9,62),(10,63),(11,64),(12,65),(13,53),(27,90),(28,91),(29,79),(30,80),(31,81),(32,82),(33,83),(34,84),(35,85),(36,86),(37,87),(38,88),(39,89),(40,73),(41,74),(42,75),(43,76),(44,77),(45,78),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72)]])
65 conjugacy classes
| class | 1 | 2 | 7A | 7B | 7C | 13A | ··· | 13L | 26A | ··· | 26L | 91A | ··· | 91AJ |
| order | 1 | 2 | 7 | 7 | 7 | 13 | ··· | 13 | 26 | ··· | 26 | 91 | ··· | 91 |
| size | 1 | 7 | 2 | 2 | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C13 | C26 | D7 | C13×D7 |
| kernel | C13×D7 | C91 | D7 | C7 | C13 | C1 |
| # reps | 1 | 1 | 12 | 12 | 3 | 36 |
Matrix representation of C13×D7 ►in GL2(𝔽547) generated by
| 509 | 0 |
| 0 | 509 |
| 0 | 1 |
| 546 | 179 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(547))| [509,0,0,509],[0,546,1,179],[0,1,1,0] >;
C13×D7 in GAP, Magma, Sage, TeX
C_{13}\times D_7 % in TeX
G:=Group("C13xD7"); // GroupNames label
G:=SmallGroup(182,1);
// by ID
G=gap.SmallGroup(182,1);
# by ID
G:=PCGroup([3,-2,-13,-7,1406]);
// Polycyclic
G:=Group<a,b,c|a^13=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export