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## G = C4.2+ 1+4order 128 = 27

### 13rd 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 — C4.2+ 1+4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C22.29C24 — C4.2+ 1+4
 Lower central C1 — C2 — C2×C4 — C4.2+ 1+4
 Upper central C1 — C22 — C2×C4○D4 — C4.2+ 1+4
 Jennings C1 — C2 — C2 — C2×C4 — C4.2+ 1+4

Generators and relations for C4.2+ 1+4
G = < a,b,c,d,e | a4=c2=e2=1, b4=a2, d2=ab2, dbd-1=ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a-1b3, be=eb, dcd-1=ece=a2c, ede=ab2d >

Subgroups: 516 in 211 conjugacy classes, 84 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C22⋊C8, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C4.4D4, C41D4, C22×C8, C2×M4(2), C2×D8, C2×SD16, C22×D4, C2×C4○D4, (C22×C8)⋊C2, C22⋊D8, D4⋊D4, C22⋊SD16, C88D4, C87D4, C8⋊D4, C82D4, C22.D8, C23.46D4, C23.19D4, C23.20D4, C22.29C24, C22.31C24, C4.2+ 1+4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, C233D4, D4○D8, D4○SD16, C4.2+ 1+4

Character table of C4.2+ 1+4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A 8B 8C 8D 8E 8F size 1 1 1 1 4 4 4 8 8 8 2 2 4 4 4 8 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 -2 -2 -2 2 -2 0 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 -2 -2 2 2 2 0 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 -2 -2 -2 -2 2 0 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 -2 -2 2 -2 -2 0 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 4 -4 0 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 -4 4 0 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 complex lifted from D4○SD16 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 complex lifted from D4○SD16

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

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

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

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

Matrix representation of C4.2+ 1+4 in GL8(𝔽17)

 1 2 0 0 0 0 0 0 16 16 0 0 0 0 0 0 0 1 0 16 0 0 0 0 16 16 1 0 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 1
,
 0 10 0 0 0 0 0 0 12 7 0 0 0 0 0 0 0 5 12 12 0 0 0 0 12 12 5 12 0 0 0 0 0 0 0 0 5 0 13 13 0 0 0 0 7 0 13 15 0 0 0 0 11 9 8 8 0 0 0 0 4 8 4 4
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 16 0 0 0 0 0 16 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 13 0 16 0 0 0 0 0 15 0 0 16
,
 16 0 0 15 0 0 0 0 0 0 1 1 0 0 0 0 16 16 0 16 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 16 2 0 0 0 0 0 0 16 1 0 0 0 0 0 0 16 13 16 16 0 0 0 0 6 15 2 1
,
 1 0 15 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 15 0 0 0 0 0 0 0 16 0 0 0 0 0 0 12 4 16 16 0 0 0 0 0 2 0 1

`G:=sub<GL(8,GF(17))| [1,16,0,16,0,0,0,0,2,16,1,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,12,0,12,0,0,0,0,10,7,5,12,0,0,0,0,0,0,12,5,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,5,7,11,4,0,0,0,0,0,0,9,8,0,0,0,0,13,13,8,4,0,0,0,0,13,15,8,4],[1,0,1,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,13,15,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[16,0,16,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,15,1,16,1,0,0,0,0,0,0,0,0,16,16,16,6,0,0,0,0,2,1,13,15,0,0,0,0,0,0,16,2,0,0,0,0,0,0,16,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,15,1,16,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,15,16,4,2,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,1] >;`

C4.2+ 1+4 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(128,1930);`
`// by ID`

`G=gap.SmallGroup(128,1930);`
`# by ID`

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

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

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