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## G = 2+ 1+4⋊4C4order 128 = 27

### 3rd 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⋊4C4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2.C25 — 2+ 1+4⋊4C4
 Lower central C1 — C2 — C2×C4 — 2+ 1+4⋊4C4
 Upper central C1 — C4 — C22×C4 — 2+ 1+4⋊4C4
 Jennings C1 — C2 — C2 — C22×C4 — 2+ 1+4⋊4C4

Generators and relations for 2+ 1+44C4
G = < a,b,c,d,e | a4=b2=e4=1, c2=d2=a2, bab=a-1, ebe-1=ac=ca, ad=da, eae-1=a2d, bc=cb, bd=db, dcd-1=a2c, ece-1=a2bd, ede-1=a >

Subgroups: 556 in 274 conjugacy classes, 66 normal (18 characteristic)
C1, C2, C2 [×9], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×15], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×28], D4 [×4], D4 [×28], Q8 [×4], Q8 [×8], C23, C23 [×2], C23 [×6], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], M4(2) [×2], C22×C4, C22×C4 [×2], C22×C4 [×6], C2×D4, C2×D4 [×4], C2×D4 [×20], C2×Q8, C2×Q8 [×2], C2×Q8 [×6], C4○D4 [×8], C4○D4 [×36], C22⋊C8 [×2], C23⋊C4 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C4≀C2 [×4], C42⋊C2 [×2], C22×C8, C2×M4(2), C2×C4○D4, C2×C4○D4 [×2], C2×C4○D4 [×6], 2+ 1+4 [×2], 2+ 1+4 [×4], 2- 1+4 [×2], 2- 1+4 [×2], (C22×C8)⋊C2, C23.C23, C23.24D4 [×2], C42⋊C22 [×2], C2.C25, 2+ 1+44C4
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, 2+ 1+44C4

Smallest permutation representation of 2+ 1+44C4
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 2)(3 4)(5 6)(7 8)(9 12)(10 11)(13 28)(14 27)(15 26)(16 25)(17 18)(19 20)(21 32)(22 31)(23 30)(24 29)
(1 7 3 5)(2 8 4 6)(9 20 11 18)(10 17 12 19)(13 29 15 31)(14 30 16 32)(21 27 23 25)(22 28 24 26)
(1 12 3 10)(2 9 4 11)(5 19 7 17)(6 20 8 18)(13 30 15 32)(14 31 16 29)(21 28 23 26)(22 25 24 27)
(1 29 10 30)(2 14 9 13)(3 31 12 32)(4 16 11 15)(5 27 19 26)(6 22 18 23)(7 25 17 28)(8 24 20 21)```

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E ··· 2J 4A 4B 4C 4D 4E 4F ··· 4K 4L 4M 4N 4O 8A 8B 8C 8D 8E 8F order 1 2 2 2 2 2 ··· 2 4 4 4 4 4 4 ··· 4 4 4 4 4 8 8 8 8 8 8 size 1 1 2 2 2 4 ··· 4 1 1 2 2 2 4 ··· 4 8 8 8 8 4 4 4 4 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 D4 D4 D4 D4 2+ 1+4⋊4C4 kernel 2+ 1+4⋊4C4 (C22×C8)⋊C2 C23.C23 C23.24D4 C42⋊C22 C2.C25 2+ 1+4 2- 1+4 C22×C4 C2×D4 C2×Q8 C4○D4 C1 # reps 1 1 1 2 2 1 4 4 2 4 2 4 4

Matrix representation of 2+ 1+44C4 in GL4(𝔽17) generated by

 16 0 0 8 4 0 13 1 0 13 0 4 4 0 0 1
,
 1 0 0 9 0 0 13 16 0 4 0 13 0 0 0 16
,
 1 0 15 0 13 0 4 16 1 0 16 0 0 1 4 0
,
 13 0 8 0 0 0 1 13 0 0 4 0 0 13 1 0
,
 8 2 15 2 0 14 5 12 5 2 15 0 5 14 5 14
`G:=sub<GL(4,GF(17))| [16,4,0,4,0,0,13,0,0,13,0,0,8,1,4,1],[1,0,0,0,0,0,4,0,0,13,0,0,9,16,13,16],[1,13,1,0,0,0,0,1,15,4,16,4,0,16,0,0],[13,0,0,0,0,0,0,13,8,1,4,1,0,13,0,0],[8,0,5,5,2,14,2,14,15,5,15,5,2,12,0,14] >;`

2+ 1+44C4 in GAP, Magma, Sage, TeX

`2_+^{1+4}\rtimes_4C_4`
`% in TeX`

`G:=Group("ES+(2,2):4C4");`
`// GroupNames label`

`G:=SmallGroup(128,526);`
`// by ID`

`G=gap.SmallGroup(128,526);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,2019,1018,521,248,1411,124]);`
`// 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,e*b*e^-1=a*c=c*a,a*d=d*a,e*a*e^-1=a^2*d,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*c,e*c*e^-1=a^2*b*d,e*d*e^-1=a>;`
`// generators/relations`

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