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G = C23.318C24order 128 = 27

35th central stem extension by C23 of C24

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C23.318C24
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — C22×C22⋊C4 — C23.318C24
 Lower central C1 — C23 — C23.318C24
 Upper central C1 — C23 — C23.318C24
 Jennings C1 — C23 — C23.318C24

Generators and relations for C23.318C24
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=g2=1, e2=b, ab=ba, ac=ca, ede-1=gdg=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 884 in 400 conjugacy classes, 112 normal (82 characteristic)
C1, C2 [×7], C2 [×10], C4 [×14], C22 [×7], C22 [×4], C22 [×58], C2×C4 [×6], C2×C4 [×42], D4 [×16], C23, C23 [×12], C23 [×54], C42 [×2], C22⋊C4 [×4], C22⋊C4 [×27], C4⋊C4 [×8], C22×C4 [×11], C22×C4 [×15], C2×D4 [×4], C2×D4 [×14], C24 [×4], C24 [×12], C2.C42 [×6], C2×C42, C2×C22⋊C4 [×16], C2×C22⋊C4 [×4], C2×C4⋊C4 [×5], C4×D4 [×4], C22≀C2 [×4], C22.D4 [×4], C23×C4 [×3], C22×D4 [×3], C25, C243C4, C23.7Q8, C23.8Q8, C23.23D4 [×2], C24.C22 [×2], C23.10D4 [×2], C23.11D4, C23.4Q8, C22×C22⋊C4, C2×C4×D4, C2×C22≀C2, C2×C22.D4, C23.318C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×6], C24, C22.D4 [×4], C22×D4 [×2], C2×C4○D4 [×3], 2+ 1+4 [×2], C2×C22.D4, C22.19C24, C22.32C24, D42, D45D4 [×2], C22.45C24, C23.318C24

Smallest permutation representation of C23.318C24
On 32 points
Generators in S32
```(1 7)(2 8)(3 5)(4 6)(9 30)(10 31)(11 32)(12 29)(13 21)(14 22)(15 23)(16 24)(17 27)(18 28)(19 25)(20 26)
(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 15)(2 16)(3 13)(4 14)(5 21)(6 22)(7 23)(8 24)(9 27)(10 28)(11 25)(12 26)(17 30)(18 31)(19 32)(20 29)
(1 28)(2 19)(3 26)(4 17)(5 20)(6 27)(7 18)(8 25)(9 22)(10 15)(11 24)(12 13)(14 30)(16 32)(21 29)(23 31)
(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 3)(2 14)(4 16)(5 7)(6 24)(8 22)(9 27)(11 25)(13 15)(17 30)(19 32)(21 23)
(1 13)(2 14)(3 15)(4 16)(5 23)(6 24)(7 21)(8 22)(9 19)(10 20)(11 17)(12 18)(25 30)(26 31)(27 32)(28 29)```

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L ··· 2Q 4A 4B 4C 4D 4E ··· 4P 4Q 4R 4S 4T order 1 2 ··· 2 2 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 4 4 4 4 size 1 1 ··· 1 2 2 2 2 4 ··· 4 2 2 2 2 4 ··· 4 8 8 8 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 C4○D4 2+ 1+4 kernel C23.318C24 C24⋊3C4 C23.7Q8 C23.8Q8 C23.23D4 C24.C22 C23.10D4 C23.11D4 C23.4Q8 C22×C22⋊C4 C2×C4×D4 C2×C22≀C2 C2×C22.D4 C22⋊C4 C2×D4 C23 C22 # reps 1 1 1 1 2 2 2 1 1 1 1 1 1 4 4 12 2

Matrix representation of C23.318C24 in GL6(𝔽5)

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

`G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,4,1,0,0,0,0,0,0,4,0,0,0,0,0,4,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,1,0,0,0,0,0,3,0,0,0,0,0,0,1,3,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[4,2,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[4,2,0,0,0,0,0,1,0,0,0,0,0,0,1,3,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;`

C23.318C24 in GAP, Magma, Sage, TeX

`C_2^3._{318}C_2^4`
`% in TeX`

`G:=Group("C2^3.318C2^4");`
`// GroupNames label`

`G:=SmallGroup(128,1150);`
`// by ID`

`G=gap.SmallGroup(128,1150);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,253,232,758,723,675]);`
`// Polycyclic`

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

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