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## G = 2- 1+4⋊2C4order 128 = 27

### 1st 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 — C2×C4 — 2- 1+4⋊2C4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C22×Q8 — C2×2- 1+4 — 2- 1+4⋊2C4
 Lower central C1 — C2 — C2×C4 — 2- 1+4⋊2C4
 Upper central C1 — C22 — C22×C4 — 2- 1+4⋊2C4
 Jennings C1 — C2 — C2 — C22×C4 — 2- 1+4⋊2C4

Generators and relations for 2- 1+42C4
G = < a,b,c,d,e | a4=b2=e4=1, c2=d2=a2, bab=a-1, ac=ca, ad=da, eae-1=abc, bc=cb, bd=db, be=eb, dcd-1=ece-1=a2c, ede-1=a2bd >

Subgroups: 524 in 276 conjugacy classes, 68 normal (18 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×11], C22 [×3], C22 [×10], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×37], D4 [×4], D4 [×18], Q8 [×4], Q8 [×18], C23, C23 [×2], C42 [×2], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4 [×2], C22×C4 [×8], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×6], C2×Q8 [×22], C4○D4 [×8], C4○D4 [×36], C2.C42 [×2], C4.10D4 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C4≀C2 [×4], C2×C42, C2×C4⋊C4, C2×M4(2) [×2], C22×Q8, C22×Q8 [×2], C2×C4○D4 [×2], C2×C4○D4 [×4], 2- 1+4 [×4], 2- 1+4 [×6], C23.67C23, C2×C4.10D4, C23.36D4 [×2], C2×C4≀C2 [×2], C2×2- 1+4, 2- 1+42C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×12], C23, C22⋊C4 [×12], C22×C4, C2×D4 [×6], C2×C22⋊C4 [×3], C22≀C2 [×4], C243C4, D4.8D4, D4.10D4, 2- 1+42C4

Smallest permutation representation of 2- 1+42C4
On 32 points
Generators in S32
```(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 25)(2 28)(3 27)(4 26)(5 19)(6 18)(7 17)(8 20)(9 30)(10 29)(11 32)(12 31)(13 22)(14 21)(15 24)(16 23)
(1 14 3 16)(2 15 4 13)(5 30 7 32)(6 31 8 29)(9 17 11 19)(10 18 12 20)(21 27 23 25)(22 28 24 26)
(1 31 3 29)(2 32 4 30)(5 15 7 13)(6 16 8 14)(9 28 11 26)(10 25 12 27)(17 22 19 24)(18 23 20 21)
(1 12 25 31)(2 7 26 19)(3 10 27 29)(4 5 28 17)(6 14 18 21)(8 16 20 23)(9 13 32 24)(11 15 30 22)```

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E ··· 4P 4Q 4R 8A 8B 8C 8D order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 8 8 8 8 size 1 1 1 1 2 2 4 4 4 4 2 2 2 2 4 ··· 4 8 8 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 D4 D4 D4 D4 D4.8D4 D4.10D4 kernel 2- 1+4⋊2C4 C23.67C23 C2×C4.10D4 C23.36D4 C2×C4≀C2 C2×2- 1+4 2- 1+4 C22×C4 C2×D4 C2×Q8 C4○D4 C2 C2 # reps 1 1 1 2 2 1 8 2 2 4 4 2 2

Matrix representation of 2- 1+42C4 in GL6(𝔽17)

 1 1 0 0 0 0 0 16 0 0 0 0 0 0 16 0 2 0 0 0 0 0 1 16 0 0 16 0 1 0 0 0 16 1 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 16 0 1 0 0 0 16 0 0 1
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 4 0 0 0 0 0 4 13 0 0 0 0 0 0 4 0 0 0 4 0 0 13
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 2 0 0 0 0 16 1 0 0 0 0 0 1 0 16 0 0 16 1 1 0
,
 13 0 0 0 0 0 8 4 0 0 0 0 0 0 1 15 0 0 0 0 1 16 0 0 0 0 7 16 0 4 0 0 7 16 4 0

`G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,1,16,0,0,0,0,0,0,16,0,16,16,0,0,0,0,0,1,0,0,2,1,1,1,0,0,0,16,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,16,16,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,4,0,4,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,16,0,16,0,0,2,1,1,1,0,0,0,0,0,1,0,0,0,0,16,0],[13,8,0,0,0,0,0,4,0,0,0,0,0,0,1,1,7,7,0,0,15,16,16,16,0,0,0,0,0,4,0,0,0,0,4,0] >;`

2- 1+42C4 in GAP, Magma, Sage, TeX

`2_-^{1+4}\rtimes_2C_4`
`% in TeX`

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

`G:=SmallGroup(128,525);`
`// by ID`

`G=gap.SmallGroup(128,525);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,352,2019,1018,521,248,1411]);`
`// Polycyclic`

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

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