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G = C23.9C24order 128 = 27

9th non-split extension by C23 of C24 acting via C24/C22=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C23.9C24
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C4○D4 — C2.C25 — C23.9C24
 Lower central C1 — C2 — C23 — C23.9C24
 Upper central C1 — C2 — C22×C4 — C23.9C24
 Jennings C1 — C2 — C23 — C23.9C24

Generators and relations for C23.9C24
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=1, g2=c, ab=ba, faf=ac=ca, ede=ad=da, ae=ea, ag=ga, ebe=bc=cb, fdf=bd=db, bf=fb, bg=gb, gdg-1=cd=dc, ce=ec, cf=fc, cg=gc, ef=fe, eg=ge, fg=gf >

Subgroups: 860 in 386 conjugacy classes, 106 normal (8 characteristic)
C1, C2, C2 [×11], C4, C4 [×3], C4 [×12], C22 [×3], C22 [×31], C2×C4 [×12], C2×C4 [×30], D4 [×46], Q8 [×10], C23, C23 [×6], C23 [×12], C42 [×3], C22⋊C4 [×6], C22⋊C4 [×12], C4⋊C4 [×3], C22×C4, C22×C4 [×6], C22×C4 [×4], C2×D4 [×9], C2×D4 [×36], C2×Q8 [×3], C2×Q8 [×6], C4○D4 [×40], C24 [×2], C23⋊C4 [×12], C42⋊C2 [×3], C22≀C2 [×12], C4⋊D4 [×6], C4.4D4 [×3], C41D4 [×3], C22×D4, C2×C4○D4 [×3], C2×C4○D4 [×6], 2+ 1+4 [×4], 2+ 1+4 [×3], 2- 1+4 [×3], C23.C23 [×3], C2≀C22 [×8], C22.29C24 [×3], C2.C25, C23.9C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], C2×C22≀C2, C23.9C24

Character table of C23.9C24

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P size 1 1 2 2 2 4 4 4 4 4 4 8 8 2 2 2 2 4 4 4 4 4 4 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 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 -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 -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 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 -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 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 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 -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 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 -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 -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 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 -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 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 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 -1 1 -1 linear of order 2 ρ17 2 2 -2 -2 2 0 0 -2 0 2 0 0 0 2 2 -2 -2 -2 0 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 -2 -2 0 2 0 0 0 2 0 0 -2 2 -2 2 0 0 0 0 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 -2 2 -2 2 0 0 2 0 0 0 0 -2 2 2 -2 0 -2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 -2 -2 2 0 0 -2 0 -2 0 0 0 -2 -2 2 2 2 0 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 2 2 -2 -2 0 -2 0 0 0 -2 0 0 -2 2 -2 2 0 0 0 0 2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ22 2 2 -2 2 -2 -2 0 0 2 0 0 0 0 2 -2 -2 2 0 -2 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ23 2 2 2 -2 -2 0 2 0 0 0 -2 0 0 2 -2 2 -2 0 0 0 0 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ24 2 2 -2 -2 2 0 0 2 0 2 0 0 0 -2 -2 2 2 -2 0 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ25 2 2 -2 2 -2 2 0 0 -2 0 0 0 0 2 -2 -2 2 0 2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ26 2 2 2 -2 -2 0 -2 0 0 0 2 0 0 2 -2 2 -2 0 0 0 0 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ27 2 2 -2 2 -2 -2 0 0 -2 0 0 0 0 -2 2 2 -2 0 2 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ28 2 2 -2 -2 2 0 0 2 0 -2 0 0 0 2 2 -2 -2 2 0 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ29 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C23.9C24
On 16 points - transitive group 16T239
Generators in S16
```(1 11)(2 12)(3 9)(4 10)(5 16)(6 13)(7 14)(8 15)
(5 7)(6 8)(13 15)(14 16)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(2 4)(5 16)(6 15)(7 14)(8 13)(10 12)
(1 16)(2 13)(3 14)(4 15)(5 11)(6 12)(7 9)(8 10)
(5 7)(6 8)(9 11)(10 12)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)```

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

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

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

`G:=TransitiveGroup(16,239);`

On 16 points - transitive group 16T313
Generators in S16
```(5 7)(6 8)(13 15)(14 16)
(9 11)(10 12)(13 15)(14 16)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(2 4)(6 8)(10 12)(13 15)
(1 11)(2 12)(3 9)(4 10)(5 13)(6 14)(7 15)(8 16)
(1 7)(2 8)(3 5)(4 6)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)```

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

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

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

`G:=TransitiveGroup(16,313);`

Matrix representation of C23.9C24 in GL8(ℤ)

 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 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 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
,
 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 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0
,
 0 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 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 0 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 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 1 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0

`G:=sub<GL(8,Integers())| [0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[-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,-1,0,0,0,0,0,0,0,0,-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],[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,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0],[0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;`

C23.9C24 in GAP, Magma, Sage, TeX

`C_2^3._9C_2^4`
`% in TeX`

`G:=Group("C2^3.9C2^4");`
`// GroupNames label`

`G:=SmallGroup(128,1759);`
`// by ID`

`G=gap.SmallGroup(128,1759);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,2019,248,718,2028]);`
`// Polycyclic`

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

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