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## G = He7order 343 = 73

### Heisenberg group

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

 Derived series C1 — C7 — He7
 Chief series C1 — C7 — C72 — He7
 Lower central C1 — C7 — He7
 Upper central C1 — C7 — He7
 Jennings C1 — C7 — He7

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

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