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## G = C4.C22≀C2order 128 = 27

### 2nd non-split extension by C4 of C22≀C2 acting via C22≀C2/C2×D4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C4.C22≀C2
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — Q8×C23 — C4.C22≀C2
 Lower central C1 — C2 — C23 — C4.C22≀C2
 Upper central C1 — C22 — C23×C4 — C4.C22≀C2
 Jennings C1 — C2 — C2 — C22×C4 — C4.C22≀C2

Generators and relations for C4.C22≀C2
G = < a,b,c,d,e,f | a4=c2=d2=e2=1, b2=a2, f2=a, bab-1=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=a2bd, cd=dc, fcf-1=ce=ec, de=ed, fdf-1=a2d, ef=fe >

Subgroups: 500 in 282 conjugacy classes, 72 normal (10 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, Q8, C23, C23, C23, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C22⋊C8, C4.10D4, C2×M4(2), C23×C4, C23×C4, C22×Q8, C22×Q8, C24.4C4, C2×C4.10D4, Q8×C23, C4.C22≀C2
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4.10D4, C2×C22⋊C4, C22≀C2, C243C4, C2×C4.10D4, C4.C22≀C2

Smallest permutation representation of C4.C22≀C2
On 32 points
Generators in S32
```(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)
(1 12 5 16)(2 20 6 24)(3 10 7 14)(4 18 8 22)(9 32 13 28)(11 30 15 26)(17 29 21 25)(19 27 23 31)
(1 5)(2 28)(3 7)(4 30)(6 32)(8 26)(9 20)(10 14)(11 22)(12 16)(13 24)(15 18)(17 21)(19 23)(25 29)(27 31)
(1 27)(2 32)(3 29)(4 26)(5 31)(6 28)(7 25)(8 30)(9 24)(10 21)(11 18)(12 23)(13 20)(14 17)(15 22)(16 19)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 24)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 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)| (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32), (1,12,5,16)(2,20,6,24)(3,10,7,14)(4,18,8,22)(9,32,13,28)(11,30,15,26)(17,29,21,25)(19,27,23,31), (1,5)(2,28)(3,7)(4,30)(6,32)(8,26)(9,20)(10,14)(11,22)(12,16)(13,24)(15,18)(17,21)(19,23)(25,29)(27,31), (1,27)(2,32)(3,29)(4,26)(5,31)(6,28)(7,25)(8,30)(9,24)(10,21)(11,18)(12,23)(13,20)(14,17)(15,22)(16,19), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23), (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( (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32), (1,12,5,16)(2,20,6,24)(3,10,7,14)(4,18,8,22)(9,32,13,28)(11,30,15,26)(17,29,21,25)(19,27,23,31), (1,5)(2,28)(3,7)(4,30)(6,32)(8,26)(9,20)(10,14)(11,22)(12,16)(13,24)(15,18)(17,21)(19,23)(25,29)(27,31), (1,27)(2,32)(3,29)(4,26)(5,31)(6,28)(7,25)(8,30)(9,24)(10,21)(11,18)(12,23)(13,20)(14,17)(15,22)(16,19), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23), (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([[(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32)], [(1,12,5,16),(2,20,6,24),(3,10,7,14),(4,18,8,22),(9,32,13,28),(11,30,15,26),(17,29,21,25),(19,27,23,31)], [(1,5),(2,28),(3,7),(4,30),(6,32),(8,26),(9,20),(10,14),(11,22),(12,16),(13,24),(15,18),(17,21),(19,23),(25,29),(27,31)], [(1,27),(2,32),(3,29),(4,26),(5,31),(6,28),(7,25),(8,30),(9,24),(10,21),(11,18),(12,23),(13,20),(14,17),(15,22),(16,19)], [(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,24),(10,17),(11,18),(12,19),(13,20),(14,21),(15,22),(16,23)], [(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 ··· 2I 4A 4B 4C 4D 4E ··· 4N 8A ··· 8H order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 ··· 2 2 2 2 2 4 ··· 4 8 ··· 8

32 irreducible representations

 dim 1 1 1 1 1 1 2 2 4 type + + + + + + - image C1 C2 C2 C2 C4 C4 D4 D4 C4.10D4 kernel C4.C22≀C2 C24.4C4 C2×C4.10D4 Q8×C23 C23×C4 C22×Q8 C22×C4 C2×Q8 C22 # reps 1 2 4 1 4 4 4 8 4

Matrix representation of C4.C22≀C2 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 10 0 0 0 0 10 1 0 0 0 0 0 0 1 7 0 0 0 0 7 16
,
 16 0 0 0 0 0 16 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
,
 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
,
 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
,
 13 8 0 0 0 0 13 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 16 0 0 0

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

C4.C22≀C2 in GAP, Magma, Sage, TeX

`C_4.C_2^2\wr C_2`
`% in TeX`

`G:=Group("C4.C2^2wrC2");`
`// GroupNames label`

`G:=SmallGroup(128,516);`
`// by ID`

`G=gap.SmallGroup(128,516);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,456,422,2019,2028,124]);`
`// Polycyclic`

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

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