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## G = 2+ 1+4order 32 = 25

### Extraspecial group

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

Aliases: 2+ 1+4, D4D4, Q8Q8, D44C22, C2.4C24, C4.9C23, Q84C22, C232C22, C22.2C23, C4○D43C2, (C2×D4)⋊6C2, (C2×C4)⋊2C22, 2-Sylow(Omega+(4,3)), SmallGroup(32,49)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — 2+ 1+4
 Chief series C1 — C2 — C22 — C23 — C2×D4 — 2+ 1+4
 Lower central C1 — C2 — 2+ 1+4
 Upper central C1 — C2 — 2+ 1+4
 Jennings C1 — C2 — 2+ 1+4

Generators and relations for 2+ 1+4
G = < a,b,c,d | a4=b2=d2=1, c2=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c >

Subgroups: 110 in 83 conjugacy classes, 68 normal (3 characteristic)
C1, C2, C2 [×9], C4 [×6], C22 [×9], C22 [×6], C2×C4 [×9], D4 [×18], Q8 [×2], C23 [×6], C2×D4 [×9], C4○D4 [×6], 2+ 1+4
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C24, 2+ 1+4

Character table of 2+ 1+4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 4A 4B 4C 4D 4E 4F size 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 -1 -1 1 linear of order 2 ρ3 1 1 1 -1 -1 1 -1 -1 -1 1 -1 1 1 -1 1 -1 1 linear of order 2 ρ4 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ5 1 1 -1 -1 -1 -1 -1 1 1 -1 1 1 1 1 -1 -1 1 linear of order 2 ρ6 1 1 -1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 linear of order 2 ρ7 1 1 -1 -1 -1 -1 1 -1 -1 1 1 -1 -1 1 1 1 1 linear of order 2 ρ8 1 1 -1 1 1 -1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 linear of order 2 ρ9 1 1 1 -1 1 -1 -1 -1 1 -1 -1 1 -1 1 1 1 -1 linear of order 2 ρ10 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 1 -1 -1 1 -1 linear of order 2 ρ11 1 1 1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 -1 linear of order 2 ρ12 1 1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 -1 linear of order 2 ρ13 1 1 -1 1 -1 1 -1 1 -1 -1 -1 -1 1 1 1 1 -1 linear of order 2 ρ14 1 1 -1 -1 1 1 -1 1 -1 1 1 1 -1 -1 -1 1 -1 linear of order 2 ρ15 1 1 -1 1 -1 1 1 -1 1 1 -1 1 -1 1 -1 -1 -1 linear of order 2 ρ16 1 1 -1 -1 1 1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 linear of order 2 ρ17 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of 2+ 1+4
On 8 points - transitive group 8T22
Generators in S8
```(1 2 3 4)(5 6 7 8)
(1 4)(2 3)(5 8)(6 7)
(1 5 3 7)(2 6 4 8)
(1 7)(2 8)(3 5)(4 6)```

`G:=sub<Sym(8)| (1,2,3,4)(5,6,7,8), (1,4)(2,3)(5,8)(6,7), (1,5,3,7)(2,6,4,8), (1,7)(2,8)(3,5)(4,6)>;`

`G:=Group( (1,2,3,4)(5,6,7,8), (1,4)(2,3)(5,8)(6,7), (1,5,3,7)(2,6,4,8), (1,7)(2,8)(3,5)(4,6) );`

`G=PermutationGroup([(1,2,3,4),(5,6,7,8)], [(1,4),(2,3),(5,8),(6,7)], [(1,5,3,7),(2,6,4,8)], [(1,7),(2,8),(3,5),(4,6)])`

`G:=TransitiveGroup(8,22);`

On 16 points - transitive group 16T23
Generators in S16
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 14)(2 13)(3 16)(4 15)(5 12)(6 11)(7 10)(8 9)
(1 8 3 6)(2 5 4 7)(9 16 11 14)(10 13 12 15)
(1 6)(2 7)(3 8)(4 5)(9 16)(10 13)(11 14)(12 15)```

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

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

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

`G:=TransitiveGroup(16,23);`

Polynomial with Galois group 2+ 1+4 over ℚ
actionf(x)Disc(f)
8T22x8-16x6-12x5+73x4+96x3-56x2-124x-41216·32·54·72·312

Matrix representation of 2+ 1+4 in GL4(ℤ) generated by

 0 0 0 -1 0 0 1 0 0 -1 0 0 1 0 0 0
,
 1 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 -1
,
 0 1 0 0 -1 0 0 0 0 0 0 1 0 0 -1 0
,
 0 1 0 0 1 0 0 0 0 0 0 -1 0 0 -1 0
`G:=sub<GL(4,Integers())| [0,0,0,1,0,0,-1,0,0,1,0,0,-1,0,0,0],[1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,-1],[0,-1,0,0,1,0,0,0,0,0,0,-1,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,0,-1,0,0,-1,0] >;`

2+ 1+4 in GAP, Magma, Sage, TeX

`2_+^{1+4}`
`% in TeX`

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

`G:=SmallGroup(32,49);`
`// by ID`

`G=gap.SmallGroup(32,49);`
`# by ID`

`G:=PCGroup([5,-2,2,2,2,-2,181,157,483]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^2=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^2*c>;`
`// generators/relations`

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