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## G = 2- 1+4order 32 = 25

### Extraspecial group

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

Aliases: 2- 1+4, D4Q8, C2.5C24, C4.10C23, D4.4C22, Q8.4C22, C22.3C23, C4○D44C2, (C2×Q8)⋊5C2, (C2×C4).8C22, Dirac group of gamma matrices, SmallGroup(32,50)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — 2- 1+4
 Chief series C1 — C2 — C22 — C2×C4 — C2×Q8 — 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=1, c2=d2=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a2c >

Subgroups: 78 in 73 conjugacy classes, 68 normal (3 characteristic)
C1, C2, C2 [×5], C4 [×10], C22 [×5], C2×C4 [×15], D4 [×10], Q8 [×10], C2×Q8 [×5], C4○D4 [×10], 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 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 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 symplectic faithful, Schur index 2

Permutation representations of 2- 1+4
On 16 points - transitive group 16T20
Generators in S16
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 16)(2 15)(3 14)(4 13)(5 11)(6 10)(7 9)(8 12)
(1 11 3 9)(2 12 4 10)(5 14 7 16)(6 15 8 13)
(1 4 3 2)(5 8 7 6)(9 10 11 12)(13 14 15 16)```

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

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

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

`G:=TransitiveGroup(16,20);`

2- 1+4 is a maximal subgroup of
C2.C25  D4.A4  2- 1+4⋊C5
Q8.D2p: D4.8D4  D4.10D4  D4○SD16  Q8○D8  Q8.15D6  Q8○D12  Q8.10D10  D4.10D10 ...
2- 1+4 is a maximal quotient of
C23.32C23  C23.33C23  C22.31C24  C22.33C24  C22.35C24  C22.36C24  C23.41C23  D46D4  Q85D4  D4×Q8  C22.46C24  C22.50C24  Q83Q8  C22.56C24  C22.57C24  C22.58C24
(C2×C4).D2p: C23.38C23  Q8.15D6  Q8○D12  Q8.10D10  D4.10D10  Q8.10D14  D4.10D14  Q8.10D22 ...

Matrix representation of 2- 1+4 in GL4(𝔽3) generated by

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

2- 1+4 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(32,50);`
`// by ID`

`G=gap.SmallGroup(32,50);`
`# by ID`

`G:=PCGroup([5,-2,2,2,2,-2,181,86,157,72,483]);`
`// Polycyclic`

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

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