direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C7×D13, C13⋊C14, C91⋊2C2, SmallGroup(182,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C7×D13 |
Generators and relations for C7×D13
G = < a,b,c | a7=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C7×D13
►On 91 points
Generators in S91
(1 84 74 64 44 32 23)(2 85 75 65 45 33 24)(3 86 76 53 46 34 25)(4 87 77 54 47 35 26)(5 88 78 55 48 36 14)(6 89 66 56 49 37 15)(7 90 67 57 50 38 16)(8 91 68 58 51 39 17)(9 79 69 59 52 27 18)(10 80 70 60 40 28 19)(11 81 71 61 41 29 20)(12 82 72 62 42 30 21)(13 83 73 63 43 31 22)
(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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 18)(15 17)(19 26)(20 25)(21 24)(22 23)(27 36)(28 35)(29 34)(30 33)(31 32)(37 39)(40 47)(41 46)(42 45)(43 44)(48 52)(49 51)(53 61)(54 60)(55 59)(56 58)(62 65)(63 64)(66 68)(69 78)(70 77)(71 76)(72 75)(73 74)(79 88)(80 87)(81 86)(82 85)(83 84)(89 91)
G:=sub<Sym(91)| (1,84,74,64,44,32,23)(2,85,75,65,45,33,24)(3,86,76,53,46,34,25)(4,87,77,54,47,35,26)(5,88,78,55,48,36,14)(6,89,66,56,49,37,15)(7,90,67,57,50,38,16)(8,91,68,58,51,39,17)(9,79,69,59,52,27,18)(10,80,70,60,40,28,19)(11,81,71,61,41,29,20)(12,82,72,62,42,30,21)(13,83,73,63,43,31,22), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,18)(15,17)(19,26)(20,25)(21,24)(22,23)(27,36)(28,35)(29,34)(30,33)(31,32)(37,39)(40,47)(41,46)(42,45)(43,44)(48,52)(49,51)(53,61)(54,60)(55,59)(56,58)(62,65)(63,64)(66,68)(69,78)(70,77)(71,76)(72,75)(73,74)(79,88)(80,87)(81,86)(82,85)(83,84)(89,91)>;
G:=Group( (1,84,74,64,44,32,23)(2,85,75,65,45,33,24)(3,86,76,53,46,34,25)(4,87,77,54,47,35,26)(5,88,78,55,48,36,14)(6,89,66,56,49,37,15)(7,90,67,57,50,38,16)(8,91,68,58,51,39,17)(9,79,69,59,52,27,18)(10,80,70,60,40,28,19)(11,81,71,61,41,29,20)(12,82,72,62,42,30,21)(13,83,73,63,43,31,22), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,18)(15,17)(19,26)(20,25)(21,24)(22,23)(27,36)(28,35)(29,34)(30,33)(31,32)(37,39)(40,47)(41,46)(42,45)(43,44)(48,52)(49,51)(53,61)(54,60)(55,59)(56,58)(62,65)(63,64)(66,68)(69,78)(70,77)(71,76)(72,75)(73,74)(79,88)(80,87)(81,86)(82,85)(83,84)(89,91) );
G=PermutationGroup([[(1,84,74,64,44,32,23),(2,85,75,65,45,33,24),(3,86,76,53,46,34,25),(4,87,77,54,47,35,26),(5,88,78,55,48,36,14),(6,89,66,56,49,37,15),(7,90,67,57,50,38,16),(8,91,68,58,51,39,17),(9,79,69,59,52,27,18),(10,80,70,60,40,28,19),(11,81,71,61,41,29,20),(12,82,72,62,42,30,21),(13,83,73,63,43,31,22)], [(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)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(14,18),(15,17),(19,26),(20,25),(21,24),(22,23),(27,36),(28,35),(29,34),(30,33),(31,32),(37,39),(40,47),(41,46),(42,45),(43,44),(48,52),(49,51),(53,61),(54,60),(55,59),(56,58),(62,65),(63,64),(66,68),(69,78),(70,77),(71,76),(72,75),(73,74),(79,88),(80,87),(81,86),(82,85),(83,84),(89,91)]])
C7×D13 is a maximal subgroup of
C91⋊C4
56 conjugacy classes
| class | 1 | 2 | 7A | ··· | 7F | 13A | ··· | 13F | 14A | ··· | 14F | 91A | ··· | 91AJ |
| order | 1 | 2 | 7 | ··· | 7 | 13 | ··· | 13 | 14 | ··· | 14 | 91 | ··· | 91 |
| size | 1 | 13 | 1 | ··· | 1 | 2 | ··· | 2 | 13 | ··· | 13 | 2 | ··· | 2 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C7 | C14 | D13 | C7×D13 |
| kernel | C7×D13 | C91 | D13 | C13 | C7 | C1 |
| # reps | 1 | 1 | 6 | 6 | 6 | 36 |
Matrix representation of C7×D13 ►in GL2(𝔽547) generated by
| 520 | 0 |
| 0 | 520 |
| 366 | 1 |
| 334 | 388 |
| 388 | 546 |
| 118 | 159 |
G:=sub<GL(2,GF(547))| [520,0,0,520],[366,334,1,388],[388,118,546,159] >;
C7×D13 in GAP, Magma, Sage, TeX
C_7\times D_{13} % in TeX
G:=Group("C7xD13"); // GroupNames label
G:=SmallGroup(182,2);
// by ID
G=gap.SmallGroup(182,2);
# by ID
G:=PCGroup([3,-2,-7,-13,1514]);
// Polycyclic
G:=Group<a,b,c|a^7=b^13=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export