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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 — C23 — 2+ 1+4⋊2C4
 Chief series C1 — C2 — C22 — C23 — C24 — C22×D4 — C2×2+ 1+4 — 2+ 1+4⋊2C4
 Lower central C1 — C2 — C23 — 2+ 1+4⋊2C4
 Upper central C1 — C22 — C24 — 2+ 1+4⋊2C4
 Jennings C1 — C2 — C24 — 2+ 1+4⋊2C4

Generators and relations for 2+ 1+42C4
G = < a,b,c,d,e | a4=b2=d2=e4=1, c2=a2, bab=a-1, ac=ca, ad=da, eae-1=a-1cd, bc=cb, ede-1=bd=db, be=eb, dcd=a2c, ece-1=a2bc >

Subgroups: 780 in 335 conjugacy classes, 68 normal (10 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C22⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C24, C24, C2.C42, C23⋊C4, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C22×D4, C22×D4, C2×C4○D4, 2+ 1+4, 2+ 1+4, C23.23D4, C2×C23⋊C4, C2×2+ 1+4, 2+ 1+42C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, C22≀C2, C243C4, C2≀C22, C23.7D4, 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 5)(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(12 16)(17 28)(18 27)(19 26)(20 25)(21 29)(22 32)(23 31)(24 30)
(1 11 3 9)(2 12 4 10)(5 13 7 15)(6 14 8 16)(17 30 19 32)(18 31 20 29)(21 27 23 25)(22 28 24 26)
(9 11)(10 12)(13 15)(14 16)(17 28)(18 25)(19 26)(20 27)(21 29)(22 30)(23 31)(24 32)
(1 19 4 23)(2 21 3 17)(5 26 6 31)(7 28 8 29)(9 22 12 20)(10 18 11 24)(13 30 14 27)(15 32 16 25)```

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J ··· 2O 4A ··· 4J 4K ··· 4P order 1 2 2 2 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 1 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8

32 irreducible representations

 dim 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + image C1 C2 C2 C2 C4 D4 D4 D4 C2≀C22 C23.7D4 kernel 2+ 1+4⋊2C4 C23.23D4 C2×C23⋊C4 C2×2+ 1+4 2+ 1+4 C22×C4 C2×D4 C24 C2 C2 # reps 1 3 3 1 8 3 6 3 2 2

Matrix representation of 2+ 1+42C4 in GL6(𝔽5)

 4 0 0 0 0 0 3 1 0 0 0 0 0 0 0 4 0 0 0 0 1 0 0 0 0 0 0 0 0 4 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 4 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 4 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 1 0 0 0 0 0 0 1 0 0
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 1 4 0 0 0 0 2 4 0 0 0 0 0 0 4 1 4 4 0 0 1 4 4 4 0 0 1 1 1 4 0 0 1 1 4 1

`G:=sub<GL(6,GF(5))| [4,3,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,0,4,0,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,2,0,0,0,0,4,4,0,0,0,0,0,0,4,1,1,1,0,0,1,4,1,1,0,0,4,4,1,4,0,0,4,4,4,1] >;`

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(128,522);`
`// by ID`

`G=gap.SmallGroup(128,522);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,2019,521,1411,2028]);`
`// 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,e*a*e^-1=a^-1*c*d,b*c=c*b,e*d*e^-1=b*d=d*b,b*e=e*b,d*c*d=a^2*c,e*c*e^-1=a^2*b*c>;`
`// generators/relations`

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