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## G = C22.53C24order 64 = 26

### 39th central stem extension by C22 of C24

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C22.53C24
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×Q8 — C22.53C24
 Lower central C1 — C22 — C22.53C24
 Upper central C1 — C22 — C22.53C24
 Jennings C1 — C22 — C22.53C24

Generators and relations for C22.53C24
G = < a,b,c,d,e,f | a2=b2=1, c2=e2=b, d2=ba=ab, f2=a, dcd-1=ac=ca, fdf-1=ad=da, ae=ea, af=fa, ece-1=bc=cb, bd=db, be=eb, bf=fb, cf=fc, de=ed, ef=fe >

Subgroups: 181 in 118 conjugacy classes, 75 normal (7 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×9], C22, C22 [×12], C2×C4, C2×C4 [×10], C2×C4 [×4], D4 [×10], Q8 [×4], C23 [×4], C42, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×6], C22×C4 [×4], C2×D4 [×6], C2×Q8 [×2], C4×D4 [×4], C4×Q8 [×2], C22.D4 [×4], C4.4D4 [×4], C41D4, C22.53C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×4], C24, C2×C4○D4 [×2], 2+ 1+4, C22.53C24

Character table of C22.53C24

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 4Q size 1 1 1 1 4 4 4 4 2 2 2 2 2 2 2 2 2 2 2 2 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 linear of order 2 ρ17 2 -2 -2 2 0 0 0 0 0 0 0 2 0 -2i -2i -2 0 2i 2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ18 2 -2 -2 2 0 0 0 0 0 0 0 -2 0 2i -2i 2 0 -2i 2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ19 2 -2 2 -2 0 0 0 0 -2i 2i 2i 0 2 0 0 0 -2 0 0 -2i 0 0 0 0 0 complex lifted from C4○D4 ρ20 2 -2 2 -2 0 0 0 0 2i -2i -2i 0 2 0 0 0 -2 0 0 2i 0 0 0 0 0 complex lifted from C4○D4 ρ21 2 -2 -2 2 0 0 0 0 0 0 0 2 0 2i 2i -2 0 -2i -2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 -2 2 -2 0 0 0 0 -2i -2i 2i 0 -2 0 0 0 2 0 0 2i 0 0 0 0 0 complex lifted from C4○D4 ρ23 2 -2 -2 2 0 0 0 0 0 0 0 -2 0 -2i 2i 2 0 2i -2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ24 2 -2 2 -2 0 0 0 0 2i 2i -2i 0 -2 0 0 0 2 0 0 -2i 0 0 0 0 0 complex lifted from C4○D4 ρ25 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from 2+ 1+4

Smallest permutation representation of C22.53C24
On 32 points
Generators in S32
```(1 9)(2 10)(3 11)(4 12)(5 29)(6 30)(7 31)(8 32)(13 25)(14 26)(15 27)(16 28)(17 24)(18 21)(19 22)(20 23)
(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 28 11 14)(2 13 12 27)(3 26 9 16)(4 15 10 25)(5 23 31 18)(6 17 32 22)(7 21 29 20)(8 19 30 24)
(1 19 3 17)(2 18 4 20)(5 15 7 13)(6 14 8 16)(9 22 11 24)(10 21 12 23)(25 29 27 31)(26 32 28 30)
(1 25 9 13)(2 26 10 14)(3 27 11 15)(4 28 12 16)(5 19 29 22)(6 20 30 23)(7 17 31 24)(8 18 32 21)```

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

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

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

Matrix representation of C22.53C24 in GL4(𝔽5) generated by

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

C22.53C24 in GAP, Magma, Sage, TeX

`C_2^2._{53}C_2^4`
`% in TeX`

`G:=Group("C2^2.53C2^4");`
`// GroupNames label`

`G:=SmallGroup(64,240);`
`// by ID`

`G=gap.SmallGroup(64,240);`
`# by ID`

`G:=PCGroup([6,-2,2,2,2,-2,2,192,217,295,650,158,297,69]);`
`// Polycyclic`

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

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