Copied to
clipboard

## G = C23.C16order 128 = 27

### The non-split extension by C23 of C16 acting via C16/C4=C4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C23.C16
 Chief series C1 — C2 — C4 — C8 — C16 — C2×C16 — C2×M5(2) — C23.C16
 Lower central C1 — C2 — C22 — C23.C16
 Upper central C1 — C8 — C2×C16 — C23.C16
 Jennings C1 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C4 — C4 — C4 — C4 — C8 — C8 — C2×C16 — C23.C16

Generators and relations for C23.C16
G = < a,b,c,d | a2=b2=c2=1, d16=c, ab=ba, ac=ca, dad-1=abc, dbd-1=bc=cb, cd=dc >

Smallest permutation representation of C23.C16
On 32 points
Generators in S32
```(2 18)(3 19)(6 22)(7 23)(10 26)(11 27)(14 30)(15 31)
(2 18)(4 20)(6 22)(8 24)(10 26)(12 28)(14 30)(16 32)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(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)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 8A 8B 8C 8D 8E 8F 8G 8H 16A ··· 16H 16I 16J 16K 16L 32A ··· 32P order 1 2 2 2 4 4 4 4 8 8 8 8 8 8 8 8 16 ··· 16 16 16 16 16 32 ··· 32 size 1 1 2 4 1 1 2 4 1 1 1 1 2 2 4 4 2 ··· 2 4 4 4 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 4 type + + + + image C1 C2 C2 C4 C4 C8 C8 C16 C16 D4 M4(2) M5(2) C23.C16 kernel C23.C16 M6(2) C2×M5(2) C2×C16 C22×C8 C2×C8 C22×C4 C2×C4 C23 C16 C8 C4 C1 # reps 1 2 1 2 2 4 4 8 8 2 2 4 4

Matrix representation of C23.C16 in GL4(𝔽97) generated by

 1 0 0 0 61 96 0 0 0 0 1 0 27 0 0 96
,
 1 0 0 0 0 1 0 0 68 0 96 0 27 0 0 96
,
 96 0 0 0 0 96 0 0 0 0 96 0 0 0 0 96
,
 68 0 95 0 72 0 36 1 2 1 29 0 13 0 70 0
`G:=sub<GL(4,GF(97))| [1,61,0,27,0,96,0,0,0,0,1,0,0,0,0,96],[1,0,68,27,0,1,0,0,0,0,96,0,0,0,0,96],[96,0,0,0,0,96,0,0,0,0,96,0,0,0,0,96],[68,72,2,13,0,0,1,0,95,36,29,70,0,1,0,0] >;`

C23.C16 in GAP, Magma, Sage, TeX

`C_2^3.C_{16}`
`% in TeX`

`G:=Group("C2^3.C16");`
`// GroupNames label`

`G:=SmallGroup(128,132);`
`// by ID`

`G=gap.SmallGroup(128,132);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,1430,1018,80,102,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^2=b^2=c^2=1,d^16=c,a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c,d*b*d^-1=b*c=c*b,c*d=d*c>;`
`// generators/relations`

Export

׿
×
𝔽