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G = 7- 1+2  order 343 = 73

Extraspecial group

p-group, metacyclic, nilpotent (class 2), monomial

Aliases: 7- 1+2, C49⋊C7, C72.C7, C7.2C72, SmallGroup(343,4)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C7 — 7- 1+2
Chief series
C1 — C7 — C72 — 7- 1+2
Lower central
C1 — C7 — 7- 1+2
Upper central
C1 — C7 — 7- 1+2
Jennings
C1 — C7 — C7 — C7 — C7 — C7 — C7 — 7- 1+2

Generators and relations for 7- 1+2
 G = < a,b | a49=b7=1, bab-1=a8 >

C1
C7
7C7
C49
C49
C49
C49
C49
C49
C49
C72
7- 1+2

Smallest permutation representation of 7- 1+2
►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)

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

55 conjugacy classes

class 1 7A···7F7G···7L49A···49AP
order17···77···749···49
size11···17···77···7

55 irreducible representations

dim1117
type+
imageC1C7C77- 1+2
kernel7- 1+2C49C72C1
# reps14266

Matrix representation of 7- 1+2 ►in GL7(𝔽197)

1640339212275105
001040000
000178000
000019100
000001640
000000114
35616119839333
,
1339212275105164
010400000
001780000
000191000
000016400
000001140
00000036

G:=sub<GL(7,GF(197))| [164,0,0,0,0,0,35,0,0,0,0,0,0,6,33,104,0,0,0,0,161,92,0,178,0,0,0,19,122,0,0,191,0,0,83,75,0,0,0,164,0,93,105,0,0,0,0,114,33],[1,0,0,0,0,0,0,33,104,0,0,0,0,0,92,0,178,0,0,0,0,122,0,0,191,0,0,0,75,0,0,0,164,0,0,105,0,0,0,0,114,0,164,0,0,0,0,0,36] >;
 

7- 1+2 in GAP, Magma, Sage, TeX 

7_-^{1+2}
 
% in TeX
 
G:=Group("ES-(7,1)");
 
// GroupNames label
 
G:=SmallGroup(343,4);
 
// by ID
 
G=gap.SmallGroup(343,4);
 
# by ID
 
G:=PCGroup([3,-7,7,-7,147,337]);
 
// Polycyclic
 
G:=Group<a,b|a^49=b^7=1,b*a*b^-1=a^8>;
 
// generators/relations
 

Export

Subgroup lattice of 7- 1+2 in TeX

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