Copied to
clipboard

## G = A4⋊C16order 192 = 26·3

### The semidirect product of A4 and C16 acting via C16/C8=C2

Aliases: A4⋊C16, C8.6S4, (C2×A4).C8, C22⋊(C3⋊C16), C23.(C3⋊C8), (C4×A4).2C4, (C8×A4).3C2, C4.4(A4⋊C4), C2.1(A4⋊C8), (C22×C8).1S3, (C22×C4).1Dic3, SmallGroup(192,186)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4 — A4⋊C16
 Chief series C1 — C22 — A4 — C2×A4 — C4×A4 — C8×A4 — A4⋊C16
 Lower central A4 — A4⋊C16
 Upper central C1 — C8

Generators and relations for A4⋊C16
G = < a,b,c,d | a2=b2=c3=d16=1, cac-1=dad-1=ab=ba, cbc-1=a, bd=db, dcd-1=c-1 >

Smallest permutation representation of A4⋊C16
On 48 points
Generators in S48
```(1 9)(3 11)(5 13)(7 15)(17 25)(19 27)(21 29)(23 31)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)
(1 24 34)(2 35 25)(3 26 36)(4 37 27)(5 28 38)(6 39 29)(7 30 40)(8 41 31)(9 32 42)(10 43 17)(11 18 44)(12 45 19)(13 20 46)(14 47 21)(15 22 48)(16 33 23)
(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)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)```

`G:=sub<Sym(48)| (1,9)(3,11)(5,13)(7,15)(17,25)(19,27)(21,29)(23,31)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (1,24,34)(2,35,25)(3,26,36)(4,37,27)(5,28,38)(6,39,29)(7,30,40)(8,41,31)(9,32,42)(10,43,17)(11,18,44)(12,45,19)(13,20,46)(14,47,21)(15,22,48)(16,33,23), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)>;`

`G:=Group( (1,9)(3,11)(5,13)(7,15)(17,25)(19,27)(21,29)(23,31)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (1,24,34)(2,35,25)(3,26,36)(4,37,27)(5,28,38)(6,39,29)(7,30,40)(8,41,31)(9,32,42)(10,43,17)(11,18,44)(12,45,19)(13,20,46)(14,47,21)(15,22,48)(16,33,23), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48) );`

`G=PermutationGroup([[(1,9),(3,11),(5,13),(7,15),(17,25),(19,27),(21,29),(23,31),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32)], [(1,24,34),(2,35,25),(3,26,36),(4,37,27),(5,28,38),(6,39,29),(7,30,40),(8,41,31),(9,32,42),(10,43,17),(11,18,44),(12,45,19),(13,20,46),(14,47,21),(15,22,48),(16,33,23)], [(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),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)]])`

40 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 16A ··· 16P 24A 24B 24C 24D order 1 2 2 2 3 4 4 4 4 6 8 8 8 8 8 8 8 8 12 12 16 ··· 16 24 24 24 24 size 1 1 3 3 8 1 1 3 3 8 1 1 1 1 3 3 3 3 8 8 6 ··· 6 8 8 8 8

40 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 3 3 3 3 type + + + - + image C1 C2 C4 C8 C16 S3 Dic3 C3⋊C8 C3⋊C16 S4 A4⋊C4 A4⋊C8 A4⋊C16 kernel A4⋊C16 C8×A4 C4×A4 C2×A4 A4 C22×C8 C22×C4 C23 C22 C8 C4 C2 C1 # reps 1 1 2 4 8 1 1 2 4 2 2 4 8

Matrix representation of A4⋊C16 in GL3(𝔽97) generated by

 96 0 0 0 96 0 1 1 1
,
 96 0 0 0 1 0 0 96 96
,
 0 1 0 96 96 95 0 0 1
,
 85 85 73 0 12 0 0 0 12
`G:=sub<GL(3,GF(97))| [96,0,1,0,96,1,0,0,1],[96,0,0,0,1,96,0,0,96],[0,96,0,1,96,0,0,95,1],[85,0,0,85,12,0,73,0,12] >;`

A4⋊C16 in GAP, Magma, Sage, TeX

`A_4\rtimes C_{16}`
`% in TeX`

`G:=Group("A4:C16");`
`// GroupNames label`

`G:=SmallGroup(192,186);`
`// by ID`

`G=gap.SmallGroup(192,186);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,14,36,58,1124,4037,285,2358,475]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^2=b^2=c^3=d^16=1,c*a*c^-1=d*a*d^-1=a*b=b*a,c*b*c^-1=a,b*d=d*b,d*c*d^-1=c^-1>;`
`// generators/relations`

Export

׿
×
𝔽