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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 — C24 — C22×D4 — 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=d2=e2=f2=g2=1, cbc=gbg=ab=ba, fcf=ac=ca, ede=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: 3556 in 2996 conjugacy classes, 2826 normal (3 characteristic)
C1, C2, C2 [×35], C4 [×28], C22 [×35], C22 [×105], C2×C4 [×210], D4 [×280], Q8 [×56], C23 [×105], C23 [×30], C22×C4 [×105], C2×D4 [×630], C2×Q8 [×70], C4○D4 [×560], C24 [×30], C22×D4 [×105], C2×C4○D4 [×210], 2+ 1+4 [×280], 2- 1+4 [×56], C2×2+ 1+4 [×35], C2.C25 [×28], 2+ 1+6
Quotients: C1, C2 [×63], C22 [×651], C23 [×1395], C24 [×651], C25 [×63], C26, 2+ 1+6

Permutation representations of 2+ 1+6
On 16 points - transitive group 16T197
Generators in S16
```(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)
(1 15)(2 16)(3 14)(4 13)(5 11)(6 12)(7 10)(8 9)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)
(1 3)(2 4)(5 8)(6 7)(9 11)(10 12)(13 16)(14 15)
(1 3)(2 4)(5 7)(6 8)(9 12)(10 11)(13 16)(14 15)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)```

`G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,15)(2,16)(3,14)(4,13)(5,11)(6,12)(7,10)(8,9), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,16)(14,15), (1,3)(2,4)(5,7)(6,8)(9,12)(10,11)(13,16)(14,15), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)>;`

`G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,15)(2,16)(3,14)(4,13)(5,11)(6,12)(7,10)(8,9), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,16)(14,15), (1,3)(2,4)(5,7)(6,8)(9,12)(10,11)(13,16)(14,15), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16) );`

`G=PermutationGroup([(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)], [(1,15),(2,16),(3,14),(4,13),(5,11),(6,12),(7,10),(8,9)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16)], [(1,3),(2,4),(5,8),(6,7),(9,11),(10,12),(13,16),(14,15)], [(1,3),(2,4),(5,7),(6,8),(9,12),(10,11),(13,16),(14,15)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)])`

`G:=TransitiveGroup(16,197);`

65 conjugacy classes

 class 1 2A 2B ··· 2AJ 4A ··· 4AB 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 35 28 1

Matrix representation of 2+ 1+6 in GL8(ℤ)

 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1
,
 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 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 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 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 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 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
,
 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,Integers())| [-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[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,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0],[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],[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(128,2326);`
`// by ID`

`G=gap.SmallGroup(128,2326);`
`# by ID`

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

`G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=g^2=1,c*b*c=g*b*g=a*b=b*a,f*c*f=a*c=c*a,e*d*e=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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