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## G = C23.10C24order 128 = 27

### 10th 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 — C23.10C24
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C4○D4 — C2.C25 — C23.10C24
 Lower central C1 — C2 — C23 — C23.10C24
 Upper central C1 — C2 — C22×C4 — C23.10C24
 Jennings C1 — C2 — C23 — C23.10C24

Generators and relations for C23.10C24
G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=1, d2=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: 684 in 358 conjugacy classes, 106 normal (8 characteristic)
C1, C2, C2 [×9], C4, C4 [×3], C4 [×14], C22 [×3], C22 [×17], C2×C4 [×12], C2×C4 [×36], D4 [×30], Q8 [×14], C23, C23 [×6], C23 [×4], C42 [×3], C22⋊C4 [×6], C22⋊C4 [×12], C4⋊C4 [×15], C22×C4, C22×C4 [×8], C22×C4 [×4], C2×D4 [×9], C2×D4 [×18], C2×Q8 [×3], C2×Q8 [×12], C4○D4 [×40], C23⋊C4 [×12], C42⋊C2 [×3], C22⋊Q8 [×6], C22.D4 [×12], C4.4D4 [×3], C4⋊Q8 [×3], C22×Q8, C2×C4○D4 [×3], C2×C4○D4 [×6], 2+ 1+4 [×4], 2+ 1+4 [×3], 2- 1+4 [×3], C23.C23 [×3], C23.7D4 [×8], C23.38C23 [×3], C2.C25, C23.10C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], C2×C22≀C2, C23.10C24

Character table of C23.10C24

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 4Q 4R size 1 1 2 2 2 4 4 4 4 4 4 2 2 2 2 4 4 4 4 4 4 8 8 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 -2 0 0 -2 0 0 -2 2 2 -2 0 0 2 0 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 -2 2 0 0 -2 0 -2 0 -2 -2 2 2 0 2 0 0 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 -2 -2 0 -2 0 0 0 2 2 -2 2 -2 2 0 0 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 -2 2 -2 2 0 0 2 0 0 -2 2 2 -2 0 0 -2 0 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 2 -2 -2 2 0 0 -2 0 2 0 2 2 -2 -2 0 -2 0 0 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ22 2 2 -2 2 -2 2 0 0 -2 0 0 2 -2 -2 2 0 0 2 0 -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 2 -2 2 -2 -2 0 0 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ24 2 2 -2 2 -2 -2 0 0 2 0 0 2 -2 -2 2 0 0 -2 0 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ25 2 2 2 -2 -2 0 -2 0 0 0 -2 -2 2 -2 2 2 0 0 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ26 2 2 -2 -2 2 0 0 2 0 -2 0 2 2 -2 -2 0 2 0 0 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ27 2 2 2 -2 -2 0 2 0 0 0 2 -2 2 -2 2 -2 0 0 -2 0 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 -2 -2 2 2 0 -2 0 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 symplectic faithful, Schur index 2

Smallest permutation representation of C23.10C24
On 32 points
Generators in S32
```(1 9)(2 10)(3 11)(4 12)(5 22)(6 23)(7 24)(8 21)(13 25)(14 26)(15 27)(16 28)(17 31)(18 32)(19 29)(20 30)
(1 27)(2 28)(3 25)(4 26)(5 30)(6 31)(7 32)(8 29)(9 15)(10 16)(11 13)(12 14)(17 23)(18 24)(19 21)(20 22)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(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)
(1 31)(2 18)(3 29)(4 20)(5 16)(6 25)(7 14)(8 27)(9 17)(10 32)(11 19)(12 30)(13 23)(15 21)(22 28)(24 26)
(2 28)(4 26)(5 30)(7 32)(9 11)(10 14)(12 16)(13 15)(17 19)(18 22)(20 24)(21 23)
(1 8 3 6)(2 7 4 5)(9 21 11 23)(10 24 12 22)(13 17 15 19)(14 20 16 18)(25 31 27 29)(26 30 28 32)```

`G:=sub<Sym(32)| (1,9)(2,10)(3,11)(4,12)(5,22)(6,23)(7,24)(8,21)(13,25)(14,26)(15,27)(16,28)(17,31)(18,32)(19,29)(20,30), (1,27)(2,28)(3,25)(4,26)(5,30)(6,31)(7,32)(8,29)(9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (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), (1,31)(2,18)(3,29)(4,20)(5,16)(6,25)(7,14)(8,27)(9,17)(10,32)(11,19)(12,30)(13,23)(15,21)(22,28)(24,26), (2,28)(4,26)(5,30)(7,32)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23), (1,8,3,6)(2,7,4,5)(9,21,11,23)(10,24,12,22)(13,17,15,19)(14,20,16,18)(25,31,27,29)(26,30,28,32)>;`

`G:=Group( (1,9)(2,10)(3,11)(4,12)(5,22)(6,23)(7,24)(8,21)(13,25)(14,26)(15,27)(16,28)(17,31)(18,32)(19,29)(20,30), (1,27)(2,28)(3,25)(4,26)(5,30)(6,31)(7,32)(8,29)(9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (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), (1,31)(2,18)(3,29)(4,20)(5,16)(6,25)(7,14)(8,27)(9,17)(10,32)(11,19)(12,30)(13,23)(15,21)(22,28)(24,26), (2,28)(4,26)(5,30)(7,32)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23), (1,8,3,6)(2,7,4,5)(9,21,11,23)(10,24,12,22)(13,17,15,19)(14,20,16,18)(25,31,27,29)(26,30,28,32) );`

`G=PermutationGroup([(1,9),(2,10),(3,11),(4,12),(5,22),(6,23),(7,24),(8,21),(13,25),(14,26),(15,27),(16,28),(17,31),(18,32),(19,29),(20,30)], [(1,27),(2,28),(3,25),(4,26),(5,30),(6,31),(7,32),(8,29),(9,15),(10,16),(11,13),(12,14),(17,23),(18,24),(19,21),(20,22)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32)], [(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)], [(1,31),(2,18),(3,29),(4,20),(5,16),(6,25),(7,14),(8,27),(9,17),(10,32),(11,19),(12,30),(13,23),(15,21),(22,28),(24,26)], [(2,28),(4,26),(5,30),(7,32),(9,11),(10,14),(12,16),(13,15),(17,19),(18,22),(20,24),(21,23)], [(1,8,3,6),(2,7,4,5),(9,21,11,23),(10,24,12,22),(13,17,15,19),(14,20,16,18),(25,31,27,29),(26,30,28,32)])`

Matrix representation of C23.10C24 in GL8(𝔽5)

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

`G:=sub<GL(8,GF(5))| [0,1,1,0,0,1,0,1,0,1,0,0,0,0,0,0,1,4,0,0,0,4,0,4,0,3,0,4,0,1,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,1,1,0,2,1,2,1,1,0,1,0,4,4,4,4,0,0,4,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,1],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,2,0,0,4,2,0,2,0,4,0,0,0,1,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,1,4,0,0,0,4,0,4,0,3,0,4,0,1,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,2,3,2,1,1,1,1,3,0,3,0,2,2,2,2,0,0,2,3,0,0,0,0,0,0,4,3,0,0,0,0,0,0,0,0,0,0,2,3,0,0,0,0,0,0,4,3,0,0,0,0,2,3,0,0,0,0,0,0,4,3,0,0],[1,0,0,1,2,0,3,2,0,1,0,1,4,0,1,4,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,4],[3,0,0,0,0,1,0,1,0,3,0,0,0,2,0,2,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2] >;`

C23.10C24 in GAP, Magma, Sage, TeX

`C_2^3._{10}C_2^4`
`% in TeX`

`G:=Group("C2^3.10C2^4");`
`// GroupNames label`

`G:=SmallGroup(128,1760);`
`// by ID`

`G=gap.SmallGroup(128,1760);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,2019,248,718,2028]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=1,d^2=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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