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## G = 2- 1+6order 128 = 27

### Extraspecial group

p-group, metabelian, nilpotent (class 2), monomial, rational

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — 2- 1+6
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C22×Q8 — C2×2- 1+4 — 2- 1+6
 Lower central C1 — C2 — 2- 1+6
 Upper central C1 — C2 — 2- 1+6
 Jennings C1 — C2 — 2- 1+6

Generators and relations for 2- 1+6
G = < a,b,c,d,e,f,g | a2=b2=c2=f2=g2=1, d2=e2=a, cbc=gbg=ab=ba, fcf=ac=ca, ede-1=ad=da, ae=ea, af=fa, ag=ga, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cg=gc, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 3060 in 2898 conjugacy classes, 2826 normal (3 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, D4, Q8, C23, C22×C4, C2×D4, C2×Q8, C4○D4, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×2- 1+4, C2.C25, 2- 1+6
Quotients: C1, C2, C22, C23, C24, C25, C26, 2- 1+6

Smallest permutation representation of 2- 1+6
On 32 points
Generators in S32
```(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(1 26)(2 27)(3 28)(4 25)(5 31)(6 32)(7 29)(8 30)(9 14)(10 15)(11 16)(12 13)(17 22)(18 23)(19 24)(20 21)
(1 22)(2 23)(3 24)(4 21)(5 11)(6 12)(7 9)(8 10)(13 30)(14 31)(15 32)(16 29)(17 28)(18 25)(19 26)(20 27)
(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)
(1 8 3 6)(2 7 4 5)(9 21 11 23)(10 24 12 22)(13 17 15 19)(14 20 16 18)(25 31 27 29)(26 30 28 32)
(1 9)(2 10)(3 11)(4 12)(5 22)(6 23)(7 24)(8 21)(13 25)(14 26)(15 27)(16 28)(17 31)(18 32)(19 29)(20 30)
(1 27)(2 28)(3 25)(4 26)(5 30)(6 31)(7 32)(8 29)(9 15)(10 16)(11 13)(12 14)(17 23)(18 24)(19 21)(20 22)```

`G:=sub<Sym(32)| (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (1,26)(2,27)(3,28)(4,25)(5,31)(6,32)(7,29)(8,30)(9,14)(10,15)(11,16)(12,13)(17,22)(18,23)(19,24)(20,21), (1,22)(2,23)(3,24)(4,21)(5,11)(6,12)(7,9)(8,10)(13,30)(14,31)(15,32)(16,29)(17,28)(18,25)(19,26)(20,27), (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), (1,8,3,6)(2,7,4,5)(9,21,11,23)(10,24,12,22)(13,17,15,19)(14,20,16,18)(25,31,27,29)(26,30,28,32), (1,9)(2,10)(3,11)(4,12)(5,22)(6,23)(7,24)(8,21)(13,25)(14,26)(15,27)(16,28)(17,31)(18,32)(19,29)(20,30), (1,27)(2,28)(3,25)(4,26)(5,30)(6,31)(7,32)(8,29)(9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22)>;`

`G:=Group( (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (1,26)(2,27)(3,28)(4,25)(5,31)(6,32)(7,29)(8,30)(9,14)(10,15)(11,16)(12,13)(17,22)(18,23)(19,24)(20,21), (1,22)(2,23)(3,24)(4,21)(5,11)(6,12)(7,9)(8,10)(13,30)(14,31)(15,32)(16,29)(17,28)(18,25)(19,26)(20,27), (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), (1,8,3,6)(2,7,4,5)(9,21,11,23)(10,24,12,22)(13,17,15,19)(14,20,16,18)(25,31,27,29)(26,30,28,32), (1,9)(2,10)(3,11)(4,12)(5,22)(6,23)(7,24)(8,21)(13,25)(14,26)(15,27)(16,28)(17,31)(18,32)(19,29)(20,30), (1,27)(2,28)(3,25)(4,26)(5,30)(6,31)(7,32)(8,29)(9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22) );`

`G=PermutationGroup([[(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32)], [(1,26),(2,27),(3,28),(4,25),(5,31),(6,32),(7,29),(8,30),(9,14),(10,15),(11,16),(12,13),(17,22),(18,23),(19,24),(20,21)], [(1,22),(2,23),(3,24),(4,21),(5,11),(6,12),(7,9),(8,10),(13,30),(14,31),(15,32),(16,29),(17,28),(18,25),(19,26),(20,27)], [(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)], [(1,8,3,6),(2,7,4,5),(9,21,11,23),(10,24,12,22),(13,17,15,19),(14,20,16,18),(25,31,27,29),(26,30,28,32)], [(1,9),(2,10),(3,11),(4,12),(5,22),(6,23),(7,24),(8,21),(13,25),(14,26),(15,27),(16,28),(17,31),(18,32),(19,29),(20,30)], [(1,27),(2,28),(3,25),(4,26),(5,30),(6,31),(7,32),(8,29),(9,15),(10,16),(11,13),(12,14),(17,23),(18,24),(19,21),(20,22)]])`

65 conjugacy classes

 class 1 2A 2B ··· 2AB 4A ··· 4AJ order 1 2 2 ··· 2 4 ··· 4 size 1 1 2 ··· 2 2 ··· 2

65 irreducible representations

 dim 1 1 1 8 type + + + - image C1 C2 C2 2- 1+6 kernel 2- 1+6 C2×2- 1+4 C2.C25 C1 # reps 1 27 36 1

Matrix representation of 2- 1+6 in GL8(𝔽5)

 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4
,
 0 0 3 0 4 0 0 0 0 0 0 2 0 1 0 0 3 0 0 0 0 0 4 0 0 2 0 0 0 0 0 1 3 0 0 0 0 0 2 0 0 2 0 0 0 0 0 3 0 0 3 0 2 0 0 0 0 0 0 2 0 3 0 0
,
 0 0 0 3 0 0 0 0 0 0 3 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 3 2 0 0 0 0 0 3 0 0 0 0 3 0 2 0 0 0 0 3 0 2 0 0 0
,
 0 0 0 3 0 4 0 0 0 0 3 0 4 0 0 0 0 3 0 0 0 0 0 4 3 0 0 0 0 0 4 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0
,
 0 0 1 0 3 0 0 0 0 0 0 1 0 3 0 0 1 0 0 0 0 0 3 0 0 1 0 0 0 0 0 3 1 0 0 0 0 0 4 0 0 1 0 0 0 0 0 4 0 0 1 0 4 0 0 0 0 0 0 1 0 4 0 0
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

`G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,0,3,0,3,0,0,0,0,0,0,2,0,2,0,0,3,0,0,0,0,0,3,0,0,2,0,0,0,0,0,2,4,0,0,0,0,0,2,0,0,1,0,0,0,0,0,3,0,0,4,0,2,0,0,0,0,0,0,1,0,3,0,0],[0,0,0,2,0,2,0,0,0,0,2,0,2,0,0,0,0,3,0,0,0,0,0,3,3,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0],[0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,4,0,0,0,0,0,2,4,0,0,0,0,0,2,0,0,0,0,4,0,2,0,0,0,0,4,0,2,0,0,0],[0,0,1,0,1,0,0,0,0,0,0,1,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,3,0,0,0,0,0,4,0,0,3,0,0,0,0,0,4,0,0,3,0,4,0,0,0,0,0,0,3,0,4,0,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;`

2- 1+6 in GAP, Magma, Sage, TeX

`2_-^{1+6}`
`% in TeX`

`G:=Group("ES-(2,3)");`
`// GroupNames label`

`G:=SmallGroup(128,2327);`
`// by ID`

`G=gap.SmallGroup(128,2327);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,2,2,-2,925,352,521,248,1411,4037]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=g^2=1,d^2=e^2=a,c*b*c=g*b*g=a*b=b*a,f*c*f=a*c=c*a,e*d*e^-1=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*g=g*c,d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;`
`// generators/relations`

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