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G = C72order 49 = 72

Elementary abelian group of type [7,7]

direct product, p-group, elementary abelian, monomial

Aliases: C72, SmallGroup(49,2)

Series: Derived Chief Lower central Upper central Jennings

C1 — C72
C1C7 — C72
C1 — C72
C1 — C72
C1 — C72

Generators and relations for C72
 G = < a,b | a7=b7=1, ab=ba >


Smallest permutation representation of C72
Regular action on 49 points
Generators in S49
(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)
(1 49 20 27 41 32 13)(2 43 21 28 42 33 14)(3 44 15 22 36 34 8)(4 45 16 23 37 35 9)(5 46 17 24 38 29 10)(6 47 18 25 39 30 11)(7 48 19 26 40 31 12)

G:=sub<Sym(49)| (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), (1,49,20,27,41,32,13)(2,43,21,28,42,33,14)(3,44,15,22,36,34,8)(4,45,16,23,37,35,9)(5,46,17,24,38,29,10)(6,47,18,25,39,30,11)(7,48,19,26,40,31,12)>;

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), (1,49,20,27,41,32,13)(2,43,21,28,42,33,14)(3,44,15,22,36,34,8)(4,45,16,23,37,35,9)(5,46,17,24,38,29,10)(6,47,18,25,39,30,11)(7,48,19,26,40,31,12) );

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)], [(1,49,20,27,41,32,13),(2,43,21,28,42,33,14),(3,44,15,22,36,34,8),(4,45,16,23,37,35,9),(5,46,17,24,38,29,10),(6,47,18,25,39,30,11),(7,48,19,26,40,31,12)])

49 conjugacy classes

class 1 7A···7AV
order17···7
size11···1

49 irreducible representations

dim11
type+
imageC1C7
kernelC72C7
# reps148

Matrix representation of C72 in GL2(𝔽29) generated by

250
01
,
70
023
G:=sub<GL(2,GF(29))| [25,0,0,1],[7,0,0,23] >;

C72 in GAP, Magma, Sage, TeX

C_7^2
% in TeX

G:=Group("C7^2");
// GroupNames label

G:=SmallGroup(49,2);
// by ID

G=gap.SmallGroup(49,2);
# by ID

G:=PCGroup([2,-7,7]:ExponentLimit:=1);
// Polycyclic

G:=Group<a,b|a^7=b^7=1,a*b=b*a>;
// generators/relations

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