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## G = C22.11C24order 64 = 26

### 7th central extension by C22 of C24

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C22.11C24
 Chief series C1 — C2 — C22 — C23 — C24 — C22×D4 — C22.11C24
 Lower central C1 — C2 — C22.11C24
 Upper central C1 — C22 — C22.11C24
 Jennings C1 — C22 — C22.11C24

Generators and relations for C22.11C24
G = < a,b,c,d,e,f | a2=b2=d2=e2=f2=1, c2=b, ab=ba, dcd=fcf=ac=ca, ede=ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ce=ec, df=fd, ef=fe >

Subgroups: 257 in 169 conjugacy classes, 121 normal (7 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C2×C22⋊C4, C42⋊C2, C4×D4, C22×D4, C22.11C24
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, C23×C4, 2+ 1+4, C22.11C24

Permutation representations of C22.11C24
On 16 points - transitive group 16T68
Generators in S16
```(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 6)(2 10)(3 8)(4 12)(5 14)(7 16)(9 15)(11 13)
(1 3)(2 4)(5 10)(6 11)(7 12)(8 9)(13 15)(14 16)
(1 3)(2 14)(4 16)(5 10)(6 8)(7 12)(9 11)(13 15)```

`G:=sub<Sym(16)| (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,6)(2,10)(3,8)(4,12)(5,14)(7,16)(9,15)(11,13), (1,3)(2,4)(5,10)(6,11)(7,12)(8,9)(13,15)(14,16), (1,3)(2,14)(4,16)(5,10)(6,8)(7,12)(9,11)(13,15)>;`

`G:=Group( (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,6)(2,10)(3,8)(4,12)(5,14)(7,16)(9,15)(11,13), (1,3)(2,4)(5,10)(6,11)(7,12)(8,9)(13,15)(14,16), (1,3)(2,14)(4,16)(5,10)(6,8)(7,12)(9,11)(13,15) );`

`G=PermutationGroup([[(1,15),(2,16),(3,13),(4,14),(5,12),(6,9),(7,10),(8,11)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,6),(2,10),(3,8),(4,12),(5,14),(7,16),(9,15),(11,13)], [(1,3),(2,4),(5,10),(6,11),(7,12),(8,9),(13,15),(14,16)], [(1,3),(2,14),(4,16),(5,10),(6,8),(7,12),(9,11),(13,15)]])`

`G:=TransitiveGroup(16,68);`

34 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2M 4A ··· 4T order 1 2 2 2 2 ··· 2 4 ··· 4 size 1 1 1 1 2 ··· 2 2 ··· 2

34 irreducible representations

 dim 1 1 1 1 1 1 4 type + + + + + + image C1 C2 C2 C2 C2 C4 2+ 1+4 kernel C22.11C24 C2×C22⋊C4 C42⋊C2 C4×D4 C22×D4 C2×D4 C2 # reps 1 4 2 8 1 16 2

Matrix representation of C22.11C24 in GL5(𝔽5)

 1 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4
,
 4 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 3 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0
,
 4 0 0 0 0 0 4 3 0 0 0 0 1 0 0 0 0 0 1 2 0 0 0 0 4
,
 4 0 0 0 0 0 1 0 0 0 0 4 4 0 0 0 0 0 1 0 0 0 0 4 4
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 4 0 0 0 0 0 4

`G:=sub<GL(5,GF(5))| [1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[3,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0],[4,0,0,0,0,0,4,0,0,0,0,3,1,0,0,0,0,0,1,0,0,0,0,2,4],[4,0,0,0,0,0,1,4,0,0,0,0,4,0,0,0,0,0,1,4,0,0,0,0,4],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4] >;`

C22.11C24 in GAP, Magma, Sage, TeX

`C_2^2._{11}C_2^4`
`% in TeX`

`G:=Group("C2^2.11C2^4");`
`// GroupNames label`

`G:=SmallGroup(64,199);`
`// by ID`

`G=gap.SmallGroup(64,199);`
`# by ID`

`G:=PCGroup([6,-2,2,2,2,-2,2,192,217,188,579]);`
`// Polycyclic`

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

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