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## G = C4.142+ 1+4order 128 = 27

### 14th non-split extension by C4 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C4.142+ 1+4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C22.29C24 — C4.142+ 1+4
 Lower central C1 — C2 — C2×C4 — C4.142+ 1+4
 Upper central C1 — C22 — C2×C4○D4 — C4.142+ 1+4
 Jennings C1 — C2 — C2 — C2×C4 — C4.142+ 1+4

Generators and relations for C4.142+ 1+4
G = < a,b,c,d,e | a4=c2=1, b4=e2=a2, d2=ab2, dbd-1=ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=ab3, be=eb, dcd-1=ece-1=a2c, ede-1=a-1b2d >

Subgroups: 588 in 221 conjugacy classes, 84 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C22⋊C8, D4⋊C4, C4.Q8, C2.D8, C42⋊C2, C22≀C2, C4⋊D4, C4.4D4, C41D4, C22×C8, C2×M4(2), C2×D8, C22×D4, C2×C4○D4, (C22×C8)⋊C2, C22⋊D8, C87D4, C82D4, C23.19D4, C22.29C24, C4.142+ 1+4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, C233D4, D4○D8, C4.142+ 1+4

Character table of C4.142+ 1+4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D 8E 8F size 1 1 1 1 4 4 4 8 8 8 8 2 2 4 4 4 8 8 8 8 4 4 4 4 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 1 1 1 1 1 -1 1 1 -1 1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 1 1 -1 1 -1 -1 1 1 1 1 1 -1 -1 1 -1 -1 -1 1 1 -1 1 -1 -1 1 linear of order 2 ρ6 1 1 1 1 -1 1 -1 1 -1 -1 -1 1 1 -1 -1 1 1 1 1 -1 1 -1 1 -1 -1 1 linear of order 2 ρ7 1 1 1 1 -1 1 -1 1 1 1 -1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 -1 1 1 -1 linear of order 2 ρ8 1 1 1 1 -1 1 -1 -1 -1 -1 1 1 1 -1 -1 1 -1 1 1 1 -1 1 -1 1 1 -1 linear of order 2 ρ9 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ10 1 1 1 1 1 -1 -1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ11 1 1 1 1 1 -1 -1 1 -1 1 -1 1 1 1 -1 -1 -1 1 -1 1 1 1 1 1 -1 -1 linear of order 2 ρ12 1 1 1 1 1 -1 -1 -1 1 -1 1 1 1 1 -1 -1 1 -1 1 -1 1 1 1 1 -1 -1 linear of order 2 ρ13 1 1 1 1 -1 -1 1 1 -1 1 1 1 1 -1 1 -1 -1 -1 1 -1 -1 1 -1 1 -1 1 linear of order 2 ρ14 1 1 1 1 -1 -1 1 -1 1 -1 -1 1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 linear of order 2 ρ15 1 1 1 1 -1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 1 -1 1 1 1 -1 1 -1 1 -1 linear of order 2 ρ16 1 1 1 1 -1 -1 1 1 1 -1 1 1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 -1 1 -1 linear of order 2 ρ17 2 2 2 2 2 2 -2 0 0 0 0 -2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 -2 -2 -2 0 0 0 0 -2 -2 2 2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 2 -2 2 0 0 0 0 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 -2 2 2 0 0 0 0 -2 -2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 -4 4 -4 0 0 0 0 0 0 0 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from 2+ 1+4 ρ22 4 -4 4 -4 0 0 0 0 0 0 0 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from 2+ 1+4 ρ23 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2√2 0 -2√2 0 0 orthogonal lifted from D4○D8 ρ24 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2√2 0 2√2 0 0 orthogonal lifted from D4○D8 ρ25 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2√2 0 2√2 0 0 0 orthogonal lifted from D4○D8 ρ26 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2√2 0 -2√2 0 0 0 orthogonal lifted from D4○D8

Smallest permutation representation of C4.142+ 1+4
On 32 points
Generators in S32
```(1 25 5 29)(2 26 6 30)(3 27 7 31)(4 28 8 32)(9 23 13 19)(10 24 14 20)(11 17 15 21)(12 18 16 22)
(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)
(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 19 27 11)(2 10 28 18)(3 17 29 9)(4 16 30 24)(5 23 31 15)(6 14 32 22)(7 21 25 13)(8 12 26 20)
(1 9 5 13)(2 10 6 14)(3 11 7 15)(4 12 8 16)(17 31 21 27)(18 32 22 28)(19 25 23 29)(20 26 24 30)```

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

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

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

Matrix representation of C4.142+ 1+4 in GL8(𝔽17)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 15 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 16 15 0 0 0 0 0 0 1 1
,
 0 0 16 15 0 0 0 0 0 0 1 1 0 0 0 0 16 15 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 6 6 0 0 0 0 0 0 14 0 0 0 0 0 11 11 0 0 0 0 0 0 3 0 0 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16
,
 1 2 0 0 0 0 0 0 16 16 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 16 16 0 0 0 0 0 0 0 0 0 0 16 15 0 0 0 0 0 0 0 1 0 0 0 0 1 2 0 0 0 0 0 0 0 16 0 0
,
 16 15 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0

`G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1],[0,0,16,1,0,0,0,0,0,0,15,1,0,0,0,0,16,1,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,0,0,0,0,11,3,0,0,0,0,0,0,11,0,0,0,0,0,6,14,0,0,0,0,0,0,6,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,16,0,0,0,0,0,0,2,16,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,2,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,16,0,0,0,0,16,0,0,0,0,0,0,0,15,1,0,0],[16,0,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,16,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;`

C4.142+ 1+4 in GAP, Magma, Sage, TeX

`C_4._{14}2_+^{1+4}`
`% in TeX`

`G:=Group("C4.14ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(128,1931);`
`// by ID`

`G=gap.SmallGroup(128,1931);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,219,675,1018,4037,1027,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^4=c^2=1,b^4=e^2=a^2,d^2=a*b^2,d*b*d^-1=a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=a*b^3,b*e=e*b,d*c*d^-1=e*c*e^-1=a^2*c,e*d*e^-1=a^-1*b^2*d>;`
`// generators/relations`

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