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## G = 2- 1+4⋊5C4order 128 = 27

### 4th semidirect product of 2- 1+4 and C4 acting via C4/C2=C2

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — 2- 1+4⋊5C4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2.C25 — 2- 1+4⋊5C4
 Lower central C1 — C2 — C4 — 2- 1+4⋊5C4
 Upper central C1 — C4 — C2×C4○D4 — 2- 1+4⋊5C4
 Jennings C1 — C2 — C2 — C2×C4 — 2- 1+4⋊5C4

Generators and relations for 2- 1+45C4
G = < a,b,c,d,e | a4=b2=d2=e4=1, c2=a2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=a2c, ce=ec, ede-1=a2cd >

Subgroups: 636 in 369 conjugacy classes, 170 normal (13 characteristic)
C1, C2, C2 [×11], C4 [×2], C4 [×6], C4 [×9], C22, C22 [×6], C22 [×13], C8 [×4], C2×C4, C2×C4 [×15], C2×C4 [×32], D4 [×16], D4 [×22], Q8 [×8], Q8 [×6], C23 [×3], C23 [×6], C42, C42 [×3], C42 [×3], C22⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×6], M4(2) [×4], M4(2) [×6], C22×C4 [×3], C22×C4 [×9], C2×D4 [×9], C2×D4 [×18], C2×Q8, C2×Q8 [×6], C2×Q8 [×4], C4○D4 [×24], C4○D4 [×28], C4≀C2 [×16], C2×C42 [×3], C42⋊C2 [×3], C4×D4 [×3], C4×Q8, C2×M4(2) [×6], C8○D4 [×4], C2×C4○D4, C2×C4○D4 [×6], C2×C4○D4 [×4], 2+ 1+4 [×4], 2+ 1+4 [×3], 2- 1+4 [×4], 2- 1+4, C2×C4≀C2 [×6], C42⋊C22 [×6], C4×C4○D4, Q8○M4(2), C2.C25, 2- 1+45C4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], C22×C22⋊C4, 2- 1+45C4

Permutation representations of 2- 1+45C4
On 16 points - transitive group 16T207
Generators in S16
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 3)(5 7)(10 12)(13 15)
(1 10 3 12)(2 11 4 9)(5 15 7 13)(6 16 8 14)
(1 5)(2 6)(3 7)(4 8)(9 16)(10 13)(11 14)(12 15)
(1 10 3 12)(2 11 4 9)(5 7)(6 8)(13 15)(14 16)```

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

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

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

`G:=TransitiveGroup(16,207);`

44 conjugacy classes

 class 1 2A 2B ··· 2H 2I 2J 2K 2L 4A 4B 4C ··· 4M 4N ··· 4W 8A ··· 8H order 1 2 2 ··· 2 2 2 2 2 4 4 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 2 ··· 2 4 4 4 4 1 1 2 ··· 2 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 D4 D4 D4 2- 1+4⋊5C4 kernel 2- 1+4⋊5C4 C2×C4≀C2 C42⋊C22 C4×C4○D4 Q8○M4(2) C2.C25 2+ 1+4 2- 1+4 C2×D4 C2×Q8 C4○D4 C1 # reps 1 6 6 1 1 1 8 8 3 1 4 4

Matrix representation of 2- 1+45C4 in GL4(𝔽5) generated by

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

2- 1+45C4 in GAP, Magma, Sage, TeX

`2_-^{1+4}\rtimes_5C_4`
`% in TeX`

`G:=Group("ES-(2,2):5C4");`
`// GroupNames label`

`G:=SmallGroup(128,1633);`
`// by ID`

`G=gap.SmallGroup(128,1633);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,521,2804,1411,172,124]);`
`// Polycyclic`

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

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