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## G = C2.7C2≀C4order 128 = 27

### 4th central stem extension by C2 of C2≀C4

p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — C2.7C2≀C4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C23.78C23 — C2.7C2≀C4
 Lower central C1 — C2 — C23 — C22×C4 — C2.7C2≀C4
 Upper central C1 — C22 — C23 — C2×C4⋊C4 — C2.7C2≀C4
 Jennings C1 — C22 — C23 — C2×C4⋊C4 — C2.7C2≀C4

Generators and relations for C2.7C2≀C4
G = < a,b,c,d,e,f | a2=d2=e2=1, b2=c2=f4=a, cbc-1=ab=ba, ac=ca, ad=da, ae=ea, af=fa, bd=db, be=eb, fbf-1=bcde, cd=dc, ce=ec, fcf-1=cde, fdf-1=de=ed, ef=fe >

Subgroups: 180 in 70 conjugacy classes, 20 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, Q8, C23, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×Q8, C2.C42, C2.C42, C22⋊C8, C2×C4⋊C4, C2×C4⋊C4, C2×M4(2), C22×Q8, C22.M4(2), C22.C42, C23.78C23, C2.7C2≀C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, SD16, Q16, C23⋊C4, Q8⋊C4, C4≀C2, C23.31D4, C2≀C4, C42.3C4, C2.7C2≀C4

Character table of C2.7C2≀C4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 2 2 4 4 4 4 8 8 8 8 8 8 8 8 8 8 8 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 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 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 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 linear of order 2 ρ5 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 1 -1 -i i i i -i -i -i i linear of order 4 ρ6 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 1 -1 1 i i i -i -i -i i -i linear of order 4 ρ7 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 1 -1 i -i -i -i i i i -i linear of order 4 ρ8 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 1 -1 1 -i -i -i i i i -i i linear of order 4 ρ9 2 2 2 2 2 2 2 -2 2 -2 0 0 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 2 -2 2 -2 2 0 0 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 2 -2 -2 0 2 0 0 0 0 0 0 √2 0 0 √2 0 0 -√2 -√2 symplectic lifted from Q16, Schur index 2 ρ12 2 -2 2 -2 2 -2 -2 0 2 0 0 0 0 0 0 -√2 0 0 -√2 0 0 √2 √2 symplectic lifted from Q16, Schur index 2 ρ13 2 -2 2 -2 -2 2 0 2i 0 -2i 0 0 0 0 0 0 1+i -1-i 0 1-i -1+i 0 0 complex lifted from C4≀C2 ρ14 2 -2 2 -2 -2 2 0 -2i 0 2i 0 0 0 0 0 0 1-i -1+i 0 1+i -1-i 0 0 complex lifted from C4≀C2 ρ15 2 -2 2 -2 -2 2 0 -2i 0 2i 0 0 0 0 0 0 -1+i 1-i 0 -1-i 1+i 0 0 complex lifted from C4≀C2 ρ16 2 -2 2 -2 2 -2 2 0 -2 0 0 0 0 0 0 -√-2 0 0 √-2 0 0 √-2 -√-2 complex lifted from SD16 ρ17 2 -2 2 -2 2 -2 2 0 -2 0 0 0 0 0 0 √-2 0 0 -√-2 0 0 -√-2 √-2 complex lifted from SD16 ρ18 2 -2 2 -2 -2 2 0 2i 0 -2i 0 0 0 0 0 0 -1-i 1+i 0 -1+i 1-i 0 0 complex lifted from C4≀C2 ρ19 4 4 -4 -4 0 0 0 0 0 0 -2 0 0 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from C2≀C4 ρ20 4 4 -4 -4 0 0 0 0 0 0 2 0 0 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from C2≀C4 ρ21 4 4 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ22 4 -4 -4 4 0 0 0 0 0 0 0 2 0 -2 0 0 0 0 0 0 0 0 0 symplectic lifted from C42.3C4, Schur index 2 ρ23 4 -4 -4 4 0 0 0 0 0 0 0 -2 0 2 0 0 0 0 0 0 0 0 0 symplectic lifted from C42.3C4, Schur index 2

Smallest permutation representation of C2.7C2≀C4
On 32 points
Generators in S32
```(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)
(1 13 5 9)(2 8 6 4)(3 27 7 31)(10 32 14 28)(11 17 15 21)(12 26 16 30)(18 24 22 20)(19 29 23 25)
(1 27 5 31)(2 32 6 28)(3 9 7 13)(4 14 8 10)(11 23 15 19)(12 20 16 24)(17 29 21 25)(18 26 22 30)
(1 19)(2 6)(3 21)(4 8)(5 23)(7 17)(9 25)(10 14)(11 27)(12 16)(13 29)(15 31)(18 22)(20 24)(26 30)(28 32)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 29)(10 30)(11 31)(12 32)(13 25)(14 26)(15 27)(16 28)
(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)| (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (1,13,5,9)(2,8,6,4)(3,27,7,31)(10,32,14,28)(11,17,15,21)(12,26,16,30)(18,24,22,20)(19,29,23,25), (1,27,5,31)(2,32,6,28)(3,9,7,13)(4,14,8,10)(11,23,15,19)(12,20,16,24)(17,29,21,25)(18,26,22,30), (1,19)(2,6)(3,21)(4,8)(5,23)(7,17)(9,25)(10,14)(11,27)(12,16)(13,29)(15,31)(18,22)(20,24)(26,30)(28,32), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,29)(10,30)(11,31)(12,32)(13,25)(14,26)(15,27)(16,28), (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( (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (1,13,5,9)(2,8,6,4)(3,27,7,31)(10,32,14,28)(11,17,15,21)(12,26,16,30)(18,24,22,20)(19,29,23,25), (1,27,5,31)(2,32,6,28)(3,9,7,13)(4,14,8,10)(11,23,15,19)(12,20,16,24)(17,29,21,25)(18,26,22,30), (1,19)(2,6)(3,21)(4,8)(5,23)(7,17)(9,25)(10,14)(11,27)(12,16)(13,29)(15,31)(18,22)(20,24)(26,30)(28,32), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,29)(10,30)(11,31)(12,32)(13,25)(14,26)(15,27)(16,28), (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([[(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32)], [(1,13,5,9),(2,8,6,4),(3,27,7,31),(10,32,14,28),(11,17,15,21),(12,26,16,30),(18,24,22,20),(19,29,23,25)], [(1,27,5,31),(2,32,6,28),(3,9,7,13),(4,14,8,10),(11,23,15,19),(12,20,16,24),(17,29,21,25),(18,26,22,30)], [(1,19),(2,6),(3,21),(4,8),(5,23),(7,17),(9,25),(10,14),(11,27),(12,16),(13,29),(15,31),(18,22),(20,24),(26,30),(28,32)], [(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,29),(10,30),(11,31),(12,32),(13,25),(14,26),(15,27),(16,28)], [(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)]])`

Matrix representation of C2.7C2≀C4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 3 0 0 0 0 11 0 0 0 0 0 0 0 0 16 0 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 4 0 0 0 0 0 0 13 0 0 0 0 0 0 0 16 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 1 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 0 0 0 0 0 0 1
,
 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 16 0 0 0 0 0 0 16
,
 8 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 0 16 0 0 0 0 16 0 0 0 0 1 0 0 0 0 16 0 0 0

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,11,0,0,0,0,3,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[8,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16,0,0,0] >;`

C2.7C2≀C4 in GAP, Magma, Sage, TeX

`C_2._7C_2\wr C_4`
`% in TeX`

`G:=Group("C2.7C2wrC4");`
`// GroupNames label`

`G:=SmallGroup(128,86);`
`// by ID`

`G=gap.SmallGroup(128,86);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,568,422,387,520,1690,521,248,2804]);`
`// Polycyclic`

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

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