p-group, metabelian, nilpotent (class 2), monomial
Aliases: He7, 7+ 1+2, C72⋊C7, C7.1C72, 7-Sylow(SL(3,7)), SmallGroup(343,3)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for He7
G = < a,b,c | a7=b7=c7=1, cac-1=ab=ba, bc=cb >
Smallest permutation representation of He7
►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 38 21 27 13 33)(2 43 39 15 28 14 34)(3 44 40 16 22 8 35)(4 45 41 17 23 9 29)(5 46 42 18 24 10 30)(6 47 36 19 25 11 31)(7 48 37 20 26 12 32)
(1 16 24 20 2 41 36)(3 46 7 28 29 31 27)(4 6 13 44 42 48 14)(5 32 15 9 11 21 35)(8 30 12 39 23 25 38)(10 26 43 17 19 49 22)(18 37 34 45 47 33 40)
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), (1,49,38,21,27,13,33)(2,43,39,15,28,14,34)(3,44,40,16,22,8,35)(4,45,41,17,23,9,29)(5,46,42,18,24,10,30)(6,47,36,19,25,11,31)(7,48,37,20,26,12,32), (1,16,24,20,2,41,36)(3,46,7,28,29,31,27)(4,6,13,44,42,48,14)(5,32,15,9,11,21,35)(8,30,12,39,23,25,38)(10,26,43,17,19,49,22)(18,37,34,45,47,33,40)>;
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), (1,49,38,21,27,13,33)(2,43,39,15,28,14,34)(3,44,40,16,22,8,35)(4,45,41,17,23,9,29)(5,46,42,18,24,10,30)(6,47,36,19,25,11,31)(7,48,37,20,26,12,32), (1,16,24,20,2,41,36)(3,46,7,28,29,31,27)(4,6,13,44,42,48,14)(5,32,15,9,11,21,35)(8,30,12,39,23,25,38)(10,26,43,17,19,49,22)(18,37,34,45,47,33,40) );
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)], [(1,49,38,21,27,13,33),(2,43,39,15,28,14,34),(3,44,40,16,22,8,35),(4,45,41,17,23,9,29),(5,46,42,18,24,10,30),(6,47,36,19,25,11,31),(7,48,37,20,26,12,32)], [(1,16,24,20,2,41,36),(3,46,7,28,29,31,27),(4,6,13,44,42,48,14),(5,32,15,9,11,21,35),(8,30,12,39,23,25,38),(10,26,43,17,19,49,22),(18,37,34,45,47,33,40)]])
55 conjugacy classes
| class | 1 | 7A | ··· | 7F | 7G | ··· | 7BB |
| order | 1 | 7 | ··· | 7 | 7 | ··· | 7 |
| size | 1 | 1 | ··· | 1 | 7 | ··· | 7 |
55 irreducible representations
| dim | 1 | 1 | 7 |
| type | + | ||
| image | C1 | C7 | He7 |
| kernel | He7 | C72 | C1 |
| # reps | 1 | 48 | 6 |
Matrix representation of He7 ►in GL7(𝔽29)
| 0 | 22 | 14 | 26 | 23 | 13 | 13 |
| 0 | 26 | 10 | 28 | 18 | 23 | 21 |
| 0 | 28 | 12 | 27 | 26 | 11 | 25 |
| 1 | 27 | 17 | 13 | 20 | 11 | 28 |
| 0 | 8 | 0 | 0 | 3 | 3 | 23 |
| 0 | 0 | 12 | 0 | 16 | 17 | 11 |
| 0 | 0 | 0 | 18 | 19 | 18 | 16 |
| 20 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 20 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 20 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 20 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 20 |
| 12 | 1 | 0 | 0 | 0 | 0 | 0 |
| 4 | 0 | 1 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 1 | 0 | 0 | 0 |
| 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 15 | 0 | 0 | 0 | 24 | 26 | 17 |
G:=sub<GL(7,GF(29))| [0,0,0,1,0,0,0,22,26,28,27,8,0,0,14,10,12,17,0,12,0,26,28,27,13,0,0,18,23,18,26,20,3,16,19,13,23,11,11,3,17,18,13,21,25,28,23,11,16],[20,0,0,0,0,0,0,0,20,0,0,0,0,0,0,0,20,0,0,0,0,0,0,0,20,0,0,0,0,0,0,0,20,0,0,0,0,0,0,0,20,0,0,0,0,0,0,0,20],[12,4,8,6,0,0,15,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,24,0,0,0,0,1,0,26,0,0,0,0,0,1,17] >;
He7 in GAP, Magma, Sage, TeX
{\rm He}_7 % in TeX
G:=Group("He7"); // GroupNames label
G:=SmallGroup(343,3);
// by ID
G=gap.SmallGroup(343,3);
# by ID
G:=PCGroup([3,-7,7,-7,337]);
// Polycyclic
G:=Group<a,b,c|a^7=b^7=c^7=1,c*a*c^-1=a*b=b*a,b*c=c*b>;
// generators/relations
Export