direct product, abelian, monomial, 7-elementary
Aliases: C7×C14, SmallGroup(98,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C7×C14 |
| C1 — C7×C14 |
| C1 — C7×C14 |
Generators and relations for C7×C14
G = < a,b | a7=b14=1, ab=ba >
Smallest permutation representation of C7×C14
►Regular action on 98 points
Generators in S98
(1 35 77 23 59 90 54)(2 36 78 24 60 91 55)(3 37 79 25 61 92 56)(4 38 80 26 62 93 43)(5 39 81 27 63 94 44)(6 40 82 28 64 95 45)(7 41 83 15 65 96 46)(8 42 84 16 66 97 47)(9 29 71 17 67 98 48)(10 30 72 18 68 85 49)(11 31 73 19 69 86 50)(12 32 74 20 70 87 51)(13 33 75 21 57 88 52)(14 34 76 22 58 89 53)
(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 92 93 94 95 96 97 98)
G:=sub<Sym(98)| (1,35,77,23,59,90,54)(2,36,78,24,60,91,55)(3,37,79,25,61,92,56)(4,38,80,26,62,93,43)(5,39,81,27,63,94,44)(6,40,82,28,64,95,45)(7,41,83,15,65,96,46)(8,42,84,16,66,97,47)(9,29,71,17,67,98,48)(10,30,72,18,68,85,49)(11,31,73,19,69,86,50)(12,32,74,20,70,87,51)(13,33,75,21,57,88,52)(14,34,76,22,58,89,53), (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98)>;
G:=Group( (1,35,77,23,59,90,54)(2,36,78,24,60,91,55)(3,37,79,25,61,92,56)(4,38,80,26,62,93,43)(5,39,81,27,63,94,44)(6,40,82,28,64,95,45)(7,41,83,15,65,96,46)(8,42,84,16,66,97,47)(9,29,71,17,67,98,48)(10,30,72,18,68,85,49)(11,31,73,19,69,86,50)(12,32,74,20,70,87,51)(13,33,75,21,57,88,52)(14,34,76,22,58,89,53), (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98) );
G=PermutationGroup([[(1,35,77,23,59,90,54),(2,36,78,24,60,91,55),(3,37,79,25,61,92,56),(4,38,80,26,62,93,43),(5,39,81,27,63,94,44),(6,40,82,28,64,95,45),(7,41,83,15,65,96,46),(8,42,84,16,66,97,47),(9,29,71,17,67,98,48),(10,30,72,18,68,85,49),(11,31,73,19,69,86,50),(12,32,74,20,70,87,51),(13,33,75,21,57,88,52),(14,34,76,22,58,89,53)], [(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),(85,86,87,88,89,90,91,92,93,94,95,96,97,98)]])
C7×C14 is a maximal subgroup of
C7⋊Dic7
98 conjugacy classes
| class | 1 | 2 | 7A | ··· | 7AV | 14A | ··· | 14AV |
| order | 1 | 2 | 7 | ··· | 7 | 14 | ··· | 14 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
98 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C7 | C14 |
| kernel | C7×C14 | C72 | C14 | C7 |
| # reps | 1 | 1 | 48 | 48 |
Matrix representation of C7×C14 ►in GL2(𝔽29) generated by
| 1 | 0 |
| 0 | 25 |
| 9 | 0 |
| 0 | 9 |
G:=sub<GL(2,GF(29))| [1,0,0,25],[9,0,0,9] >;
C7×C14 in GAP, Magma, Sage, TeX
C_7\times C_{14} % in TeX
G:=Group("C7xC14"); // GroupNames label
G:=SmallGroup(98,5);
// by ID
G=gap.SmallGroup(98,5);
# by ID
G:=PCGroup([3,-2,-7,-7]);
// Polycyclic
G:=Group<a,b|a^7=b^14=1,a*b=b*a>;
// generators/relations
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