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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 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C2×2+ 1+4 — 2+ 1+4⋊5C4
 Lower central C1 — C2 — C4 — 2+ 1+4⋊5C4
 Upper central C1 — C22 — 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=eae-1=a-1, ac=ca, ad=da, bc=cb, bd=db, be=eb, dcd=ece-1=a2c, ede-1=cd >

Subgroups: 764 in 389 conjugacy classes, 172 normal (18 characteristic)
C1, C2 [×3], C2 [×12], C4 [×2], C4 [×6], C4 [×6], C22, C22 [×6], C22 [×30], C8 [×4], C2×C4, C2×C4 [×15], C2×C4 [×20], D4 [×18], D4 [×27], Q8 [×6], Q8, C23 [×3], C23 [×21], C42 [×3], C22⋊C4 [×3], C4⋊C4, C4⋊C4 [×3], C4⋊C4 [×3], C2×C8, C2×C8 [×3], C2×C8 [×6], M4(2) [×6], C22×C4 [×3], C22×C4 [×6], C2×D4 [×18], C2×D4 [×36], C2×Q8 [×2], C4○D4 [×20], C4○D4 [×14], C24 [×3], D4⋊C4 [×12], Q8⋊C4, Q8⋊C4 [×3], C2×C4⋊C4 [×3], C42⋊C2 [×3], C4×D4 [×3], C4×Q8, C22×C8 [×3], C2×M4(2) [×3], C8○D4 [×4], C22×D4 [×3], C22×D4 [×3], C2×C4○D4, C2×C4○D4 [×3], C2×C4○D4, 2+ 1+4 [×8], 2+ 1+4 [×4], C2×D4⋊C4 [×3], C23.24D4 [×3], C23.36D4 [×3], C23.37D4 [×3], C23.33C23, C2×C8○D4, C2×2+ 1+4, 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, D4○D8, D4○SD16, 2+ 1+45C4

Smallest permutation representation of 2+ 1+45C4
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 23)(2 22)(3 21)(4 24)(5 11)(6 10)(7 9)(8 12)(13 26)(14 25)(15 28)(16 27)(17 31)(18 30)(19 29)(20 32)
(1 9 3 11)(2 10 4 12)(5 23 7 21)(6 24 8 22)(13 32 15 30)(14 29 16 31)(17 25 19 27)(18 26 20 28)
(1 22)(2 23)(3 24)(4 21)(5 10)(6 11)(7 12)(8 9)(13 19)(14 20)(15 17)(16 18)(25 32)(26 29)(27 30)(28 31)
(1 26 22 14)(2 25 23 13)(3 28 24 16)(4 27 21 15)(5 32 12 19)(6 31 9 18)(7 30 10 17)(8 29 11 20)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J ··· 2O 4A ··· 4H 4I ··· 4R 8A 8B 8C 8D 8E ··· 8J order 1 2 2 2 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 8 8 8 8 8 ··· 8 size 1 1 1 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 2 2 2 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 D4 D4 D4○D8 D4○SD16 kernel 2+ 1+4⋊5C4 C2×D4⋊C4 C23.24D4 C23.36D4 C23.37D4 C23.33C23 C2×C8○D4 C2×2+ 1+4 2+ 1+4 C2×D4 C2×Q8 C4○D4 C2 C2 # reps 1 3 3 3 3 1 1 1 16 3 1 4 2 2

Matrix representation of 2+ 1+45C4 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 16 0 0 0 0 1 0 0 0 0 16 0 0 0 0 1 0 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 16 0 0 0 0 1 0 0 0
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0
,
 16 0 0 0 0 0 3 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 0 0 1
,
 11 13 0 0 0 0 5 6 0 0 0 0 0 0 0 0 3 14 0 0 0 0 14 14 0 0 3 14 0 0 0 0 14 14 0 0

`G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1,0,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[16,3,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[11,5,0,0,0,0,13,6,0,0,0,0,0,0,0,0,3,14,0,0,0,0,14,14,0,0,3,14,0,0,0,0,14,14,0,0] >;`

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(128,1629);`
`// by ID`

`G=gap.SmallGroup(128,1629);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,521,2804,1411,172]);`
`// 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,b*d=d*b,b*e=e*b,d*c*d=e*c*e^-1=a^2*c,e*d*e^-1=c*d>;`
`// generators/relations`

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