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G = C7×C14order 98 = 2·72

Abelian group of type [7,14]

direct product, abelian, monomial, 7-elementary

Aliases: C7×C14, SmallGroup(98,5)

Series: Derived Chief Lower central Upper central

C1 — C7×C14
C1C7C72 — 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 66 41 49 16 92 77)(2 67 42 50 17 93 78)(3 68 29 51 18 94 79)(4 69 30 52 19 95 80)(5 70 31 53 20 96 81)(6 57 32 54 21 97 82)(7 58 33 55 22 98 83)(8 59 34 56 23 85 84)(9 60 35 43 24 86 71)(10 61 36 44 25 87 72)(11 62 37 45 26 88 73)(12 63 38 46 27 89 74)(13 64 39 47 28 90 75)(14 65 40 48 15 91 76)
(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)| (1,66,41,49,16,92,77)(2,67,42,50,17,93,78)(3,68,29,51,18,94,79)(4,69,30,52,19,95,80)(5,70,31,53,20,96,81)(6,57,32,54,21,97,82)(7,58,33,55,22,98,83)(8,59,34,56,23,85,84)(9,60,35,43,24,86,71)(10,61,36,44,25,87,72)(11,62,37,45,26,88,73)(12,63,38,46,27,89,74)(13,64,39,47,28,90,75)(14,65,40,48,15,91,76), (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( (1,66,41,49,16,92,77)(2,67,42,50,17,93,78)(3,68,29,51,18,94,79)(4,69,30,52,19,95,80)(5,70,31,53,20,96,81)(6,57,32,54,21,97,82)(7,58,33,55,22,98,83)(8,59,34,56,23,85,84)(9,60,35,43,24,86,71)(10,61,36,44,25,87,72)(11,62,37,45,26,88,73)(12,63,38,46,27,89,74)(13,64,39,47,28,90,75)(14,65,40,48,15,91,76), (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([(1,66,41,49,16,92,77),(2,67,42,50,17,93,78),(3,68,29,51,18,94,79),(4,69,30,52,19,95,80),(5,70,31,53,20,96,81),(6,57,32,54,21,97,82),(7,58,33,55,22,98,83),(8,59,34,56,23,85,84),(9,60,35,43,24,86,71),(10,61,36,44,25,87,72),(11,62,37,45,26,88,73),(12,63,38,46,27,89,74),(13,64,39,47,28,90,75),(14,65,40,48,15,91,76)], [(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···7AV14A···14AV
order127···714···14
size111···11···1

98 irreducible representations

dim1111
type++
imageC1C2C7C14
kernelC7×C14C72C14C7
# reps114848

Matrix representation of C7×C14 in GL2(𝔽29) generated by

10
025
,
90
09
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

Export

Subgroup lattice of C7×C14 in TeX

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