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## G = C23.7C24order 128 = 27

### 7th 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 — C23 — C23.7C24
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C4○D4 — C2.C25 — C23.7C24
 Lower central C1 — C2 — C23 — C23.7C24
 Upper central C1 — C4 — C22×C4 — C23.7C24
 Jennings C1 — C2 — C23 — C23.7C24

Generators and relations for C23.7C24
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, cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 772 in 383 conjugacy classes, 106 normal (10 characteristic)
C1, C2, C2 [×11], C4, C4 [×3], C4 [×13], C22 [×3], C22 [×27], C2×C4 [×12], C2×C4 [×37], D4 [×42], Q8 [×10], C23, C23 [×6], C23 [×9], C42 [×3], C22⋊C4 [×6], C22⋊C4 [×12], C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×6], C22×C4 [×10], C2×D4 [×9], C2×D4 [×24], C2×Q8 [×3], C2×Q8 [×6], C4○D4 [×40], C24, C23⋊C4 [×12], C42⋊C2 [×3], C4×D4 [×6], C22≀C2 [×6], C4⋊D4 [×3], C22⋊Q8 [×3], C22.D4 [×6], C23×C4, 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 [×4], C23.7D4 [×4], C22.19C24 [×3], C2.C25, C23.7C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], C2×C22≀C2, C23.7C24

Permutation representations of C23.7C24
On 16 points - transitive group 16T223
Generators in S16
```(1 9)(2 10)(3 11)(4 12)(5 14)(6 15)(7 16)(8 13)
(1 11)(2 12)(3 9)(4 10)(5 14)(6 15)(7 16)(8 13)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 3)(2 4)(5 16)(6 13)(7 14)(8 15)(9 11)(10 12)
(1 14)(2 15)(3 16)(4 13)(5 9)(6 10)(7 11)(8 12)
(1 14)(2 15)(3 16)(4 13)(5 11)(6 12)(7 9)(8 10)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)```

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

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

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

`G:=TransitiveGroup(16,223);`

On 16 points - transitive group 16T270
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)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)
(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), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12), (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), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12), (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)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12)], [(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(16,270);`

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E ··· 2L 4A 4B 4C 4D 4E 4F ··· 4M 4N ··· 4S order 1 2 2 2 2 2 ··· 2 4 4 4 4 4 4 ··· 4 4 ··· 4 size 1 1 2 2 2 4 ··· 4 1 1 2 2 2 4 ··· 4 8 ··· 8

32 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D4 C23.7C24 kernel C23.7C24 C23.C23 C2≀C22 C23.7D4 C22.19C24 C2.C25 C22×C4 C2×D4 C2×Q8 C1 # reps 1 3 4 4 3 1 6 3 3 4

Matrix representation of C23.7C24 in GL4(𝔽5) generated by

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

C23.7C24 in GAP, Magma, Sage, TeX

`C_2^3._7C_2^4`
`% in TeX`

`G:=Group("C2^3.7C2^4");`
`// GroupNames label`

`G:=SmallGroup(128,1757);`
`// by ID`

`G=gap.SmallGroup(128,1757);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,248,718,2028]);`
`// 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,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;`
`// generators/relations`

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