Copied to
clipboard

## G = C22.26C24order 64 = 26

### 12nd 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 — C22.26C24
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C22.26C24
 Lower central C1 — C22 — C22.26C24
 Upper central C1 — C2×C4 — C22.26C24
 Jennings C1 — C22 — C22.26C24

Generators and relations for C22.26C24
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e2=b, f2=a, ab=ba, dcd=ac=ca, ad=da, ae=ea, af=fa, ece-1=bc=cb, bd=db, be=eb, bf=fb, cf=fc, de=ed, df=fd, ef=fe >

Subgroups: 233 in 155 conjugacy classes, 85 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×8], C4 [×6], C22, C22 [×2], C22 [×14], C2×C4 [×2], C2×C4 [×12], C2×C4 [×12], D4 [×20], Q8 [×4], C23, C23 [×4], C42 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4, C22×C4 [×6], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×8], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×2], C41D4, C4⋊Q8, C2×C4○D4 [×2], C22.26C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], C22.26C24

Character table of C22.26C24

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 4Q 4R size 1 1 1 1 2 2 4 4 4 4 1 1 1 1 2 2 2 2 2 2 2 2 2 2 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 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 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 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 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 -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 -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 -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 -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 -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 -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 -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 -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 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 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 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 1 -1 -1 linear of order 2 ρ17 2 -2 2 -2 -2 2 0 0 0 0 2 -2 2 -2 0 -2 0 0 0 0 0 2 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 2 0 0 0 0 0 -2 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 2 0 0 0 0 0 -2 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 -2 0 0 0 0 0 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 2 -2 -2 0 0 0 0 0 0 -2i 2i 2i -2i 0 0 0 0 2 -2i 2i 0 0 -2 0 0 0 0 complex lifted from C4○D4 ρ22 2 -2 -2 2 0 0 0 0 0 0 2i 2i -2i -2i -2 0 2i 2 0 0 0 0 -2i 0 0 0 0 0 complex lifted from C4○D4 ρ23 2 2 -2 -2 0 0 0 0 0 0 2i -2i -2i 2i 0 0 0 0 -2 -2i 2i 0 0 2 0 0 0 0 complex lifted from C4○D4 ρ24 2 -2 -2 2 0 0 0 0 0 0 -2i -2i 2i 2i -2 0 -2i 2 0 0 0 0 2i 0 0 0 0 0 complex lifted from C4○D4 ρ25 2 2 -2 -2 0 0 0 0 0 0 -2i 2i 2i -2i 0 0 0 0 -2 2i -2i 0 0 2 0 0 0 0 complex lifted from C4○D4 ρ26 2 -2 -2 2 0 0 0 0 0 0 2i 2i -2i -2i 2 0 -2i -2 0 0 0 0 2i 0 0 0 0 0 complex lifted from C4○D4 ρ27 2 2 -2 -2 0 0 0 0 0 0 2i -2i -2i 2i 0 0 0 0 2 2i -2i 0 0 -2 0 0 0 0 complex lifted from C4○D4 ρ28 2 -2 -2 2 0 0 0 0 0 0 -2i -2i 2i 2i 2 0 2i -2 0 0 0 0 -2i 0 0 0 0 0 complex lifted from C4○D4

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

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

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

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

Matrix representation of C22.26C24 in GL4(𝔽5) generated by

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

C22.26C24 in GAP, Magma, Sage, TeX

`C_2^2._{26}C_2^4`
`% in TeX`

`G:=Group("C2^2.26C2^4");`
`// GroupNames label`

`G:=SmallGroup(64,213);`
`// by ID`

`G=gap.SmallGroup(64,213);`
`# by ID`

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

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

Export

׿
×
𝔽