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## G = 2+ 1+4⋊3C4order 128 = 27

### 2nd 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⋊3C4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C22×D4 — C2×2+ 1+4 — 2+ 1+4⋊3C4
 Lower central C1 — C2 — C2×C4 — 2+ 1+4⋊3C4
 Upper central C1 — C22 — C22×C4 — 2+ 1+4⋊3C4
 Jennings C1 — C2 — C2 — C22×C4 — 2+ 1+4⋊3C4

Generators and relations for 2+ 1+43C4
G = < a,b,c,d,e | a4=b2=d2=e4=1, c2=a2, bab=eae-1=a-1, ac=ca, ad=da, bc=cb, ebe-1=bd=db, dcd=a2c, ece-1=acd, de=ed >

Subgroups: 716 in 310 conjugacy classes, 68 normal (18 characteristic)
C1, C2 [×3], C2 [×10], C4 [×4], C4 [×7], C22 [×3], C22 [×32], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×25], D4 [×4], D4 [×34], Q8 [×4], Q8 [×2], C23, C23 [×4], C23 [×22], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4 [×6], C2×D4 [×6], C2×D4 [×42], C2×Q8 [×2], C4○D4 [×8], C4○D4 [×20], C24 [×2], C24 [×2], C4.D4 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C4≀C2 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×M4(2) [×2], C22×D4, C22×D4 [×4], C2×C4○D4 [×2], C2×C4○D4 [×2], 2+ 1+4 [×4], 2+ 1+4 [×6], C24.3C22, C2×C4.D4, C23.36D4 [×2], C2×C4≀C2 [×2], C2×2+ 1+4, 2+ 1+43C4
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, D44D4, D4.9D4, 2+ 1+43C4

Smallest permutation representation of 2+ 1+43C4
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 23)(10 22)(11 21)(12 24)(13 14)(15 16)(17 18)(19 20)(25 28)(26 27)(29 30)(31 32)
(1 23 3 21)(2 24 4 22)(5 9 7 11)(6 10 8 12)(13 17 15 19)(14 18 16 20)(25 32 27 30)(26 29 28 31)
(1 6)(2 7)(3 8)(4 5)(9 24)(10 21)(11 22)(12 23)(13 31)(14 32)(15 29)(16 30)(17 28)(18 25)(19 26)(20 27)
(1 15 22 28)(2 14 23 27)(3 13 24 26)(4 16 21 25)(5 30 10 18)(6 29 11 17)(7 32 12 20)(8 31 9 19)```

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F ··· 2M 4A 4B 4C 4D 4E ··· 4L 4M 4N 8A 8B 8C 8D order 1 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 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⋊4D4 D4.9D4 kernel 2+ 1+4⋊3C4 C24.3C22 C2×C4.D4 C23.36D4 C2×C4≀C2 C2×2+ 1+4 2+ 1+4 C2×D4 C2×Q8 C4○D4 C24 C2 C2 # reps 1 1 1 2 2 1 8 4 2 4 2 2 2

Matrix representation of 2+ 1+43C4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 1 16 15 0 0 0 0 1 0 0 0 0 16 0 0 0 0 1 1 0 16
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 16 15 0 0 1 0 0 0 0 0 0 0 0 16
,
 16 0 0 0 0 0 15 1 0 0 0 0 0 0 1 1 16 15 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1 0 16 16
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 16 15 0 0 0 0 0 1
,
 13 4 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 16 0 0 0 1 0 16 0 0 1 1 16 16 0 0 0 0 0 16

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

2+ 1+43C4 in GAP, Magma, Sage, TeX

`2_+^{1+4}\rtimes_3C_4`
`% in TeX`

`G:=Group("ES+(2,2):3C4");`
`// GroupNames label`

`G:=SmallGroup(128,524);`
`// by ID`

`G=gap.SmallGroup(128,524);`
`# by ID`

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

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

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