direct product, p-group, elementary abelian, monomial
Aliases: C72, SmallGroup(49,2)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| 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 C72⋊3C3 He7 7- 1+2
C72 is a maximal quotient of He7 7- 1+2
49 conjugacy classes
| class | 1 | 7A | ··· | 7AV |
| order | 1 | 7 | ··· | 7 |
| size | 1 | 1 | ··· | 1 |
49 irreducible representations
| dim | 1 | 1 |
| type | + | |
| image | C1 | C7 |
| kernel | C72 | C7 |
| # reps | 1 | 48 |
Matrix representation of C72 ►in GL2(𝔽29) generated by
| 25 | 0 |
| 0 | 1 |
| 7 | 0 |
| 0 | 23 |
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