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## G = 7- 1+2order 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 >

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 ··· 7F 7G ··· 7L 49A ··· 49AP order 1 7 ··· 7 7 ··· 7 49 ··· 49 size 1 1 ··· 1 7 ··· 7 7 ··· 7

55 irreducible representations

 dim 1 1 1 7 type + image C1 C7 C7 7- 1+2 kernel 7- 1+2 C49 C72 C1 # reps 1 42 6 6

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

 164 0 33 92 122 75 105 0 0 104 0 0 0 0 0 0 0 178 0 0 0 0 0 0 0 191 0 0 0 0 0 0 0 164 0 0 0 0 0 0 0 114 35 6 161 19 83 93 33
,
 1 33 92 122 75 105 164 0 104 0 0 0 0 0 0 0 178 0 0 0 0 0 0 0 191 0 0 0 0 0 0 0 164 0 0 0 0 0 0 0 114 0 0 0 0 0 0 0 36

`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`

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