Copied to
clipboard

## G = C24.162C23order 128 = 27

### 2nd non-split extension by C24 of C23 acting via C23/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C24.162C23
 Chief series C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C42⋊C2 — C24.162C23
 Lower central C1 — C2 — C22 — C24.162C23
 Upper central C1 — C2×C4 — C23×C4 — C24.162C23
 Jennings C1 — C2 — C24 — C24.162C23

Generators and relations for C24.162C23
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=b, f2=c, g2=d, ab=ba, ac=ca, eae-1=ad=da, af=fa, ag=ga, bc=cb, fbf-1=bd=db, be=eb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fef-1=ade, eg=ge, fg=gf >

Subgroups: 372 in 210 conjugacy classes, 108 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×8], C4 [×12], C22 [×3], C22 [×4], C22 [×10], C2×C4 [×2], C2×C4 [×26], C2×C4 [×24], C23 [×3], C23 [×4], C23 [×2], C42 [×12], C22⋊C4 [×8], C22⋊C4 [×8], C4⋊C4 [×12], C22×C4 [×2], C22×C4 [×16], C22×C4 [×4], C24, C2×C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×6], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×8], C42⋊C2 [×8], C23×C4, C23.9D4 [×4], C2×C42⋊C2, C2×C42⋊C2 [×2], C24.162C23
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×6], Q8 [×2], C23, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C2.C42 [×8], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C2×C2.C42, C23.C23 [×2], C24.162C23

Smallest permutation representation of C24.162C23
On 32 points
Generators in S32
```(1 6)(2 25)(3 8)(4 27)(5 16)(7 14)(9 23)(10 32)(11 21)(12 30)(13 28)(15 26)(17 24)(18 29)(19 22)(20 31)
(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 26)(2 27)(3 28)(4 25)(5 14)(6 15)(7 16)(8 13)(9 29)(10 30)(11 31)(12 32)(17 22)(18 23)(19 24)(20 21)
(1 13)(2 14)(3 15)(4 16)(5 27)(6 28)(7 25)(8 26)(9 20)(10 17)(11 18)(12 19)(21 29)(22 30)(23 31)(24 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 32 26 12)(2 18 27 23)(3 22 28 17)(4 9 25 29)(5 31 14 11)(6 10 15 30)(7 21 16 20)(8 19 13 24)
(1 11 13 18)(2 12 14 19)(3 9 15 20)(4 10 16 17)(5 24 27 32)(6 21 28 29)(7 22 25 30)(8 23 26 31)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4J 4K ··· 4AH order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 4 ··· 4 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 4 type + + + + - image C1 C2 C2 C4 C4 C4 D4 Q8 C23.C23 kernel C24.162C23 C23.9D4 C2×C42⋊C2 C2×C42 C2×C4⋊C4 C42⋊C2 C22×C4 C22×C4 C2 # reps 1 4 3 4 4 16 6 2 4

Matrix representation of C24.162C23 in GL6(𝔽5)

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

`G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,1,0,0,0,0,3,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,3,0,0,0,0,2,0,0,0],[2,0,0,0,0,0,4,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3] >;`

C24.162C23 in GAP, Magma, Sage, TeX

`C_2^4._{162}C_2^3`
`% in TeX`

`G:=Group("C2^4.162C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,472);`
`// by ID`

`G=gap.SmallGroup(128,472);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,352,2019,1411]);`
`// Polycyclic`

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

׿
×
𝔽