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## G = 2+ 1+4.2C4order 128 = 27

### The non-split extension by 2+ 1+4 of 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 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2.C25 — 2+ 1+4.2C4
 Lower central C1 — C2 — C23 — 2+ 1+4.2C4
 Upper central C1 — C4 — C22×C4 — 2+ 1+4.2C4
 Jennings C1 — C2 — C2 — C22×C4 — 2+ 1+4.2C4

Generators and relations for 2+ 1+4.2C4
G = < a,b,c,d,e | a4=b2=d2=1, c2=e4=a2, bab=a-1, ac=ca, ad=da, eae-1=a-1cd, ece-1=bc=cb, bd=db, be=eb, dcd=a2c, ede-1=a2bd >

Subgroups: 540 in 274 conjugacy classes, 66 normal (8 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C22⋊C8, C4.D4, C4.10D4, C22×C8, C2×M4(2), C2×C4○D4, C2×C4○D4, 2+ 1+4, 2+ 1+4, 2- 1+4, (C22×C8)⋊C2, M4(2).8C22, C2.C25, 2+ 1+4.2C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, C22≀C2, C243C4, 2+ 1+4.2C4

Smallest permutation representation of 2+ 1+4.2C4
On 32 points
Generators in S32
```(1 25 5 29)(2 10 6 14)(3 31 7 27)(4 16 8 12)(9 23 13 19)(11 21 15 17)(18 28 22 32)(20 26 24 30)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)
(1 19 5 23)(2 16 6 12)(3 21 7 17)(4 10 8 14)(9 29 13 25)(11 31 15 27)(18 26 22 30)(20 28 24 32)
(2 28)(4 30)(6 32)(8 26)(9 13)(10 22)(11 15)(12 24)(14 18)(16 20)(17 21)(19 23)
(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)```

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E ··· 2J 4A 4B 4C 4D 4E 4F ··· 4K 8A 8B 8C 8D 8E ··· 8J order 1 2 2 2 2 2 ··· 2 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 4 4 4 4 8 ··· 8

32 irreducible representations

 dim 1 1 1 1 1 2 2 2 4 type + + + + + + + image C1 C2 C2 C2 C4 D4 D4 D4 2+ 1+4.2C4 kernel 2+ 1+4.2C4 (C22×C8)⋊C2 M4(2).8C22 C2.C25 2+ 1+4 C22×C4 C2×D4 C2×Q8 C1 # reps 1 3 3 1 8 6 3 3 4

Matrix representation of 2+ 1+4.2C4 in GL4(𝔽17) generated by

 0 0 16 2 0 0 0 1 1 15 0 0 0 16 0 0
,
 4 9 0 0 4 13 0 0 0 0 4 9 0 0 4 13
,
 0 0 16 0 0 0 0 16 1 0 0 0 0 1 0 0
,
 13 8 0 0 13 4 0 0 0 0 4 9 0 0 4 13
,
 14 8 5 9 13 5 4 14 12 8 3 9 13 3 4 12
`G:=sub<GL(4,GF(17))| [0,0,1,0,0,0,15,16,16,0,0,0,2,1,0,0],[4,4,0,0,9,13,0,0,0,0,4,4,0,0,9,13],[0,0,1,0,0,0,0,1,16,0,0,0,0,16,0,0],[13,13,0,0,8,4,0,0,0,0,4,4,0,0,9,13],[14,13,12,13,8,5,8,3,5,4,3,4,9,14,9,12] >;`

2+ 1+4.2C4 in GAP, Magma, Sage, TeX

`2_+^{1+4}._2C_4`
`% in TeX`

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

`G:=SmallGroup(128,523);`
`// by ID`

`G=gap.SmallGroup(128,523);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,2019,521,1411,2028,124]);`
`// Polycyclic`

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

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