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

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

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

C72 is a maximal subgroup of   C7⋊D7  C72⋊C3  C723C3  He7  7- 1+2
C72 is a maximal quotient of   He7  7- 1+2

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

Export

Subgroup lattice of C72 in TeX

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