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## G = C22.31C24order 64 = 26

### 17th 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 — C22.31C24
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C22.31C24
 Lower central C1 — C22 — C22.31C24
 Upper central C1 — C22 — C22.31C24
 Jennings C1 — C22 — C22.31C24

Generators and relations for C22.31C24
G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=1, f2=a, ab=ba, dcd=fcf-1=ac=ca, ede=ad=da, ae=ea, af=fa, ece=bc=cb, bd=db, be=eb, bf=fb, df=fd, ef=fe >

Subgroups: 233 in 147 conjugacy classes, 81 normal (9 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×8], C22, C22 [×2], C22 [×14], C2×C4 [×14], C2×C4 [×10], D4 [×16], Q8 [×4], C23, C23 [×4], C22⋊C4 [×8], C4⋊C4 [×8], C22×C4, C22×C4 [×6], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×8], C2×C4⋊C4, C4⋊D4 [×8], C22⋊Q8 [×4], C2×C4○D4 [×2], C22.31C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ 1+4, 2- 1+4, C22.31C24

Character table of C22.31C24

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L size 1 1 1 1 2 2 4 4 4 4 2 2 2 2 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 linear of order 2 ρ17 2 -2 2 -2 2 -2 0 0 0 0 -2 2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 -2 2 -2 -2 2 0 0 0 0 -2 -2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 2 -2 -2 2 0 0 0 0 2 2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 -2 2 -2 2 -2 0 0 0 0 2 -2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from 2+ 1+4 ρ22 4 -4 -4 4 0 0 0 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 C22.31C24
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 11)(2 12)(3 9)(4 10)(5 29)(6 30)(7 31)(8 32)(13 18)(14 19)(15 20)(16 17)(21 26)(22 27)(23 28)(24 25)
(1 4)(2 3)(5 32)(6 31)(7 30)(8 29)(9 12)(10 11)(13 19)(14 18)(15 17)(16 20)(21 24)(22 23)(25 26)(27 28)
(1 22)(2 23)(3 24)(4 21)(5 18)(6 19)(7 20)(8 17)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)
(1 14)(2 15)(3 16)(4 13)(5 28)(6 25)(7 26)(8 27)(9 17)(10 18)(11 19)(12 20)(21 31)(22 32)(23 29)(24 30)
(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)```

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

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

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

Matrix representation of C22.31C24 in GL6(ℤ)

 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 -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 1 0 0 0 0 0 0 1
,
 -1 0 0 0 0 0 0 1 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 0 1 0
,
 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 -1 0 0 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
,
 -1 0 0 0 0 0 0 -1 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 0 -1 0

`G:=sub<GL(6,Integers())| [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,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,-1,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0] >;`

C22.31C24 in GAP, Magma, Sage, TeX

`C_2^2._{31}C_2^4`
`% in TeX`

`G:=Group("C2^2.31C2^4");`
`// GroupNames label`

`G:=SmallGroup(64,218);`
`// by ID`

`G=gap.SmallGroup(64,218);`
`# by ID`

`G:=PCGroup([6,-2,2,2,2,-2,2,217,650,188,579,69]);`
`// Polycyclic`

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

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