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G = C22.57C24order 64 = 26

43rd central stem extension by C22 of C24

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C22.57C24
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C22.D4 — C22.57C24
 Lower central C1 — C22 — C22.57C24
 Upper central C1 — C22 — C22.57C24
 Jennings C1 — C22 — C22.57C24

Generators and relations for C22.57C24
G = < a,b,c,d,e,f | a2=b2=f2=1, c2=d2=e2=a, ab=ba, dcd-1=ac=ca, fdf=ad=da, ae=ea, af=fa, ece-1=bc=cb, bd=db, be=eb, bf=fb, fcf=abc, ede-1=abd, ef=fe >

Subgroups: 141 in 98 conjugacy classes, 71 normal (9 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×13], C22, C22 [×6], C2×C4, C2×C4 [×12], C2×C4 [×2], D4, Q8 [×3], C23 [×2], C42, C42 [×2], C22⋊C4 [×10], C4⋊C4 [×16], C22×C4 [×2], C2×D4, C2×Q8, C2×Q8 [×2], C22⋊Q8 [×4], C22.D4 [×2], C4.4D4, C42.C2 [×2], C422C2 [×4], C4⋊Q8 [×2], C22.57C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C24, 2+ 1+4, 2- 1+4 [×2], C22.57C24

Character table of C22.57C24

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M size 1 1 1 1 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 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 linear of order 2 ρ7 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 linear of order 2 ρ9 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 linear of order 2 ρ11 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 linear of order 2 ρ13 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 linear of order 2 ρ15 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 linear of order 2 ρ17 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from 2+ 1+4 ρ18 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from 2- 1+4, Schur index 2 ρ19 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of C22.57C24
On 32 points
Generators in S32
```(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 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 24)
(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 9 3 11)(2 12 4 10)(5 21 7 23)(6 24 8 22)(13 18 15 20)(14 17 16 19)(25 31 27 29)(26 30 28 32)
(1 21 3 23)(2 14 4 16)(5 31 7 29)(6 12 8 10)(9 20 11 18)(13 25 15 27)(17 32 19 30)(22 26 24 28)
(2 26)(4 28)(5 7)(6 17)(8 19)(9 11)(10 30)(12 32)(14 24)(16 22)(18 20)(29 31)```

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

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

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

Matrix representation of C22.57C24 in GL8(𝔽5)

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

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

C22.57C24 in GAP, Magma, Sage, TeX

`C_2^2._{57}C_2^4`
`% in TeX`

`G:=Group("C2^2.57C2^4");`
`// GroupNames label`

`G:=SmallGroup(64,244);`
`// by ID`

`G=gap.SmallGroup(64,244);`
`# by ID`

`G:=PCGroup([6,-2,2,2,2,-2,2,96,217,103,650,476,86,1347,297]);`
`// Polycyclic`

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

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