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## G = C22.56C24order 64 = 26

### 42nd 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 — C22.56C24
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C22.D4 — C22.56C24
 Lower central C1 — C22 — C22.56C24
 Upper central C1 — C22 — C22.56C24
 Jennings C1 — C22 — C22.56C24

Generators and relations for C22.56C24
G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e2=a, ab=ba, dcd=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: 181 in 110 conjugacy classes, 71 normal (7 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×11], C22, C22 [×12], C2×C4, C2×C4 [×10], C2×C4 [×4], D4 [×6], Q8 [×2], C23 [×4], C42, C22⋊C4 [×12], C4⋊C4 [×10], C22×C4 [×4], C2×D4 [×6], C2×Q8 [×2], C4⋊D4 [×4], C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×2], C42.C2, C22.56C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C24, 2+ 1+4 [×2], 2- 1+4, C22.56C24

Character table of C22.56C24

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 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 orthogonal lifted from 2+ 1+4 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 C22.56C24
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 12)(2 9)(3 10)(4 11)(5 29)(6 30)(7 31)(8 32)(13 19)(14 20)(15 17)(16 18)(21 28)(22 25)(23 26)(24 27)
(1 25)(2 23)(3 27)(4 21)(5 17)(6 16)(7 19)(8 14)(9 26)(10 24)(11 28)(12 22)(13 31)(15 29)(18 30)(20 32)
(1 20)(2 13)(3 18)(4 15)(5 26)(6 22)(7 28)(8 24)(9 19)(10 16)(11 17)(12 14)(21 31)(23 29)(25 30)(27 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)
(5 29)(6 30)(7 31)(8 32)(13 15)(14 16)(17 19)(18 20)(21 26)(22 27)(23 28)(24 25)```

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

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

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

Matrix representation of C22.56C24 in GL8(𝔽5)

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

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

C22.56C24 in GAP, Magma, Sage, TeX

`C_2^2._{56}C_2^4`
`% in TeX`

`G:=Group("C2^2.56C2^4");`
`// GroupNames label`

`G:=SmallGroup(64,243);`
`// by ID`

`G=gap.SmallGroup(64,243);`
`# by ID`

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

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