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## G = C23.C24order 128 = 27

### 3rd non-split extension by C23 of C24 acting via C24/C22=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C23.C24
 Chief series C1 — C2 — C22 — C23 — C24 — C22×D4 — C2×2+ 1+4 — C23.C24
 Lower central C1 — C2 — C22 — C23.C24
 Upper central C1 — C2 — C22×D4 — C23.C24
 Jennings C1 — C2 — C23 — C23.C24

Generators and relations for C23.C24
G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=g2=1, d2=b, ab=ba, dad-1=ac=ca, ae=ea, af=fa, ag=ga, ebe=bc=cb, bd=db, bf=fb, bg=gb, cd=dc, ce=ec, gfg=cf=fc, cg=gc, ede=acd, df=fd, dg=gd, ef=fe, eg=ge >

Subgroups: 820 in 400 conjugacy classes, 170 normal (8 characteristic)
C1, C2, C2 [×15], C4 [×4], C4 [×12], C22, C22 [×10], C22 [×29], C2×C4 [×10], C2×C4 [×32], D4 [×16], D4 [×28], Q8 [×4], C23, C23 [×16], C23 [×14], C42 [×4], C22⋊C4 [×8], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4, C22×C4 [×8], C22×C4 [×8], C2×D4 [×26], C2×D4 [×32], C2×Q8 [×2], C4○D4 [×24], C24 [×6], C23⋊C4 [×16], C2×C22⋊C4 [×8], C42⋊C2 [×4], C4×D4 [×8], C22×D4, C22×D4 [×8], C2×C4○D4 [×6], 2+ 1+4 [×8], C2×C23⋊C4 [×8], C23.C23 [×4], C22.11C24 [×2], C2×2+ 1+4, C23.C24
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], C22×C22⋊C4, C23.C24

Permutation representations of C23.C24
On 16 points - transitive group 16T199
Generators in S16
```(1 3)(2 14)(4 16)(5 7)(6 11)(8 9)(10 12)(13 15)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 2)(3 14)(4 13)(5 9)(6 12)(7 8)(10 11)(15 16)
(1 10)(2 11)(3 12)(4 9)(5 13)(6 14)(7 15)(8 16)
(5 12)(6 9)(7 10)(8 11)```

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

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

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

`G:=TransitiveGroup(16,199);`

On 16 points - transitive group 16T202
Generators in S16
```(2 3)(6 8)(9 11)(13 15)
(9 11)(10 12)(13 15)(14 16)
(1 4)(2 3)(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 10)(2 9)(3 11)(4 12)(5 14)(6 13)(7 16)(8 15)
(1 7)(2 8)(3 6)(4 5)(9 15)(10 16)(11 13)(12 14)
(5 7)(6 8)(13 15)(14 16)```

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

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

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

`G:=TransitiveGroup(16,202);`

On 16 points - transitive group 16T203
Generators in S16
```(2 16)(4 14)(6 9)(8 11)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(2 16)(3 13)(5 12)(8 11)
(1 7)(2 8)(3 5)(4 6)(9 14)(10 15)(11 16)(12 13)
(5 12)(6 9)(7 10)(8 11)```

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

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

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

`G:=TransitiveGroup(16,203);`

41 conjugacy classes

 class 1 2A 2B ··· 2L 2M 2N 2O 2P 4A 4B 4C 4D 4E ··· 4X order 1 2 2 ··· 2 2 2 2 2 4 4 4 4 4 ··· 4 size 1 1 2 ··· 2 4 4 4 4 2 2 2 2 4 ··· 4

41 irreducible representations

 dim 1 1 1 1 1 1 1 2 8 type + + + + + + + image C1 C2 C2 C2 C2 C4 C4 D4 C23.C24 kernel C23.C24 C2×C23⋊C4 C23.C23 C22.11C24 C2×2+ 1+4 C22×D4 C2×C4○D4 C2×D4 C1 # reps 1 8 4 2 1 8 8 8 1

Matrix representation of C23.C24 in GL8(ℤ)

 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 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 -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 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
,
 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
,
 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

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

C23.C24 in GAP, Magma, Sage, TeX

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

`G:=Group("C2^3.C2^4");`
`// GroupNames label`

`G:=SmallGroup(128,1615);`
`// by ID`

`G=gap.SmallGroup(128,1615);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,521,2804,2028]);`
`// Polycyclic`

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

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