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## G = C49⋊C3order 147 = 3·72

### The semidirect product of C49 and C3 acting faithfully

Aliases: C49⋊C3, C7.(C7⋊C3), SmallGroup(147,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C49 — C49⋊C3
 Chief series C1 — C7 — C49 — C49⋊C3
 Lower central C49 — C49⋊C3
 Upper central C1

Generators and relations for C49⋊C3
G = < a,b | a49=b3=1, bab-1=a18 >

Character table of C49⋊C3

 class 1 3A 3B 7A 7B 49A 49B 49C 49D 49E 49F 49G 49H 49I 49J 49K 49L 49M 49N size 1 49 49 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 ζ32 ζ3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ3 1 ζ3 ζ32 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ4 3 0 0 3 3 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1+√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 complex lifted from C7⋊C3 ρ5 3 0 0 3 3 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1-√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 complex lifted from C7⋊C3 ρ6 3 0 0 -1+√-7/2 -1-√-7/2 ζ4937+ζ4932+ζ4929 ζ4941+ζ495+ζ493 ζ4940+ζ4934+ζ4924 ζ4945+ζ4927+ζ4926 ζ4920+ζ4917+ζ4912 ζ4943+ζ4939+ζ4916 ζ4947+ζ4938+ζ4913 ζ4925+ζ4915+ζ499 ζ4930+ζ4918+ζ49 ζ4936+ζ4911+ζ492 ζ4923+ζ4922+ζ494 ζ4946+ζ4944+ζ498 ζ4933+ζ4910+ζ496 ζ4948+ζ4931+ζ4919 complex faithful ρ7 3 0 0 -1-√-7/2 -1+√-7/2 ζ4933+ζ4910+ζ496 ζ4923+ζ4922+ζ494 ζ4937+ζ4932+ζ4929 ζ4936+ζ4911+ζ492 ζ4943+ζ4939+ζ4916 ζ4941+ζ495+ζ493 ζ4930+ζ4918+ζ49 ζ4920+ζ4917+ζ4912 ζ4940+ζ4934+ζ4924 ζ4948+ζ4931+ζ4919 ζ4947+ζ4938+ζ4913 ζ4945+ζ4927+ζ4926 ζ4946+ζ4944+ζ498 ζ4925+ζ4915+ζ499 complex faithful ρ8 3 0 0 -1-√-7/2 -1+√-7/2 ζ4945+ζ4927+ζ4926 ζ4930+ζ4918+ζ49 ζ4946+ζ4944+ζ498 ζ4925+ζ4915+ζ499 ζ4923+ζ4922+ζ494 ζ4947+ζ4938+ζ4913 ζ4937+ζ4932+ζ4929 ζ4941+ζ495+ζ493 ζ4933+ζ4910+ζ496 ζ4920+ζ4917+ζ4912 ζ4940+ζ4934+ζ4924 ζ4948+ζ4931+ζ4919 ζ4936+ζ4911+ζ492 ζ4943+ζ4939+ζ4916 complex faithful ρ9 3 0 0 -1-√-7/2 -1+√-7/2 ζ4940+ζ4934+ζ4924 ζ4943+ζ4939+ζ4916 ζ4930+ζ4918+ζ49 ζ4946+ζ4944+ζ498 ζ4925+ζ4915+ζ499 ζ4920+ζ4917+ζ4912 ζ4923+ζ4922+ζ494 ζ4948+ζ4931+ζ4919 ζ4947+ζ4938+ζ4913 ζ4945+ζ4927+ζ4926 ζ4941+ζ495+ζ493 ζ4933+ζ4910+ζ496 ζ4937+ζ4932+ζ4929 ζ4936+ζ4911+ζ492 complex faithful ρ10 3 0 0 -1-√-7/2 -1+√-7/2 ζ4948+ζ4931+ζ4919 ζ4937+ζ4932+ζ4929 ζ4936+ζ4911+ζ492 ζ4943+ζ4939+ζ4916 ζ4930+ζ4918+ζ49 ζ4940+ζ4934+ζ4924 ζ4946+ζ4944+ζ498 ζ4947+ζ4938+ζ4913 ζ4945+ζ4927+ζ4926 ζ4941+ζ495+ζ493 ζ4933+ζ4910+ζ496 ζ4920+ζ4917+ζ4912 ζ4925+ζ4915+ζ499 ζ4923+ζ4922+ζ494 complex faithful ρ11 3 0 0 -1-√-7/2 -1+√-7/2 ζ4947+ζ4938+ζ4913 ζ4925+ζ4915+ζ499 ζ4923+ζ4922+ζ494 ζ4937+ζ4932+ζ4929 ζ4936+ζ4911+ζ492 ζ4948+ζ4931+ζ4919 ζ4943+ζ4939+ζ4916 ζ4945+ζ4927+ζ4926 ζ4941+ζ495+ζ493 ζ4933+ζ4910+ζ496 ζ4920+ζ4917+ζ4912 ζ4940+ζ4934+ζ4924 ζ4930+ζ4918+ζ49 ζ4946+ζ4944+ζ498 complex faithful ρ12 3 0 0 -1+√-7/2 -1-√-7/2 ζ4923+ζ4922+ζ494 ζ4948+ζ4931+ζ4919 ζ4941+ζ495+ζ493 ζ4940+ζ4934+ζ4924 ζ4945+ζ4927+ζ4926 ζ4936+ζ4911+ζ492 ζ4920+ζ4917+ζ4912 ζ4946+ζ4944+ζ498 ζ4943+ζ4939+ζ4916 ζ4937+ζ4932+ζ4929 ζ4925+ζ4915+ζ499 ζ4930+ζ4918+ζ49 ζ4947+ζ4938+ζ4913 ζ4933+ζ4910+ζ496 complex faithful ρ13 3 0 0 -1+√-7/2 -1-√-7/2 ζ4946+ζ4944+ζ498 ζ4947+ζ4938+ζ4913 ζ4933+ζ4910+ζ496 ζ4948+ζ4931+ζ4919 ζ4941+ζ495+ζ493 ζ4923+ζ4922+ζ494 ζ4940+ζ4934+ζ4924 ζ4943+ζ4939+ζ4916 ζ4937+ζ4932+ζ4929 ζ4925+ζ4915+ζ499 ζ4930+ζ4918+ζ49 ζ4936+ζ4911+ζ492 ζ4945+ζ4927+ζ4926 ζ4920+ζ4917+ζ4912 complex faithful ρ14 3 0 0 -1-√-7/2 -1+√-7/2 ζ4920+ζ4917+ζ4912 ζ4946+ζ4944+ζ498 ζ4925+ζ4915+ζ499 ζ4923+ζ4922+ζ494 ζ4937+ζ4932+ζ4929 ζ4933+ζ4910+ζ496 ζ4936+ζ4911+ζ492 ζ4940+ζ4934+ζ4924 ζ4948+ζ4931+ζ4919 ζ4947+ζ4938+ζ4913 ζ4945+ζ4927+ζ4926 ζ4941+ζ495+ζ493 ζ4943+ζ4939+ζ4916 ζ4930+ζ4918+ζ49 complex faithful ρ15 3 0 0 -1+√-7/2 -1-√-7/2 ζ4925+ζ4915+ζ499 ζ4933+ζ4910+ζ496 ζ4948+ζ4931+ζ4919 ζ4941+ζ495+ζ493 ζ4940+ζ4934+ζ4924 ζ4937+ζ4932+ζ4929 ζ4945+ζ4927+ζ4926 ζ4930+ζ4918+ζ49 ζ4936+ζ4911+ζ492 ζ4923+ζ4922+ζ494 ζ4946+ζ4944+ζ498 ζ4943+ζ4939+ζ4916 ζ4920+ζ4917+ζ4912 ζ4947+ζ4938+ζ4913 complex faithful ρ16 3 0 0 -1+√-7/2 -1-√-7/2 ζ4930+ζ4918+ζ49 ζ4920+ζ4917+ζ4912 ζ4947+ζ4938+ζ4913 ζ4933+ζ4910+ζ496 ζ4948+ζ4931+ζ4919 ζ4925+ζ4915+ζ499 ζ4941+ζ495+ζ493 ζ4936+ζ4911+ζ492 ζ4923+ζ4922+ζ494 ζ4946+ζ4944+ζ498 ζ4943+ζ4939+ζ4916 ζ4937+ζ4932+ζ4929 ζ4940+ζ4934+ζ4924 ζ4945+ζ4927+ζ4926 complex faithful ρ17 3 0 0 -1-√-7/2 -1+√-7/2 ζ4941+ζ495+ζ493 ζ4936+ζ4911+ζ492 ζ4943+ζ4939+ζ4916 ζ4930+ζ4918+ζ49 ζ4946+ζ4944+ζ498 ζ4945+ζ4927+ζ4926 ζ4925+ζ4915+ζ499 ζ4933+ζ4910+ζ496 ζ4920+ζ4917+ζ4912 ζ4940+ζ4934+ζ4924 ζ4948+ζ4931+ζ4919 ζ4947+ζ4938+ζ4913 ζ4923+ζ4922+ζ494 ζ4937+ζ4932+ζ4929 complex faithful ρ18 3 0 0 -1+√-7/2 -1-√-7/2 ζ4936+ζ4911+ζ492 ζ4940+ζ4934+ζ4924 ζ4945+ζ4927+ζ4926 ζ4920+ζ4917+ζ4912 ζ4947+ζ4938+ζ4913 ζ4930+ζ4918+ζ49 ζ4933+ζ4910+ζ496 ζ4923+ζ4922+ζ494 ζ4946+ζ4944+ζ498 ζ4943+ζ4939+ζ4916 ζ4937+ζ4932+ζ4929 ζ4925+ζ4915+ζ499 ζ4948+ζ4931+ζ4919 ζ4941+ζ495+ζ493 complex faithful ρ19 3 0 0 -1+√-7/2 -1-√-7/2 ζ4943+ζ4939+ζ4916 ζ4945+ζ4927+ζ4926 ζ4920+ζ4917+ζ4912 ζ4947+ζ4938+ζ4913 ζ4933+ζ4910+ζ496 ζ4946+ζ4944+ζ498 ζ4948+ζ4931+ζ4919 ζ4937+ζ4932+ζ4929 ζ4925+ζ4915+ζ499 ζ4930+ζ4918+ζ49 ζ4936+ζ4911+ζ492 ζ4923+ζ4922+ζ494 ζ4941+ζ495+ζ493 ζ4940+ζ4934+ζ4924 complex faithful

Smallest permutation representation of C49⋊C3
On 49 points
Generators in S49
```(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 49)
(2 31 19)(3 12 37)(4 42 6)(5 23 24)(7 34 11)(8 15 29)(9 45 47)(10 26 16)(13 18 21)(14 48 39)(17 40 44)(20 32 49)(22 43 36)(25 35 41)(27 46 28)(30 38 33)```

`G:=sub<Sym(49)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49), (2,31,19)(3,12,37)(4,42,6)(5,23,24)(7,34,11)(8,15,29)(9,45,47)(10,26,16)(13,18,21)(14,48,39)(17,40,44)(20,32,49)(22,43,36)(25,35,41)(27,46,28)(30,38,33)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49), (2,31,19)(3,12,37)(4,42,6)(5,23,24)(7,34,11)(8,15,29)(9,45,47)(10,26,16)(13,18,21)(14,48,39)(17,40,44)(20,32,49)(22,43,36)(25,35,41)(27,46,28)(30,38,33) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49)], [(2,31,19),(3,12,37),(4,42,6),(5,23,24),(7,34,11),(8,15,29),(9,45,47),(10,26,16),(13,18,21),(14,48,39),(17,40,44),(20,32,49),(22,43,36),(25,35,41),(27,46,28),(30,38,33)]])`

C49⋊C3 is a maximal subgroup of   C49⋊C6
C49⋊C3 is a maximal quotient of   C49⋊C9

Matrix representation of C49⋊C3 in GL3(𝔽883) generated by

 584 610 587 587 275 597 597 366 340
,
 3 25 364 857 135 141 364 204 745
`G:=sub<GL(3,GF(883))| [584,587,597,610,275,366,587,597,340],[3,857,364,25,135,204,364,141,745] >;`

C49⋊C3 in GAP, Magma, Sage, TeX

`C_{49}\rtimes C_3`
`% in TeX`

`G:=Group("C49:C3");`
`// GroupNames label`

`G:=SmallGroup(147,1);`
`// by ID`

`G=gap.SmallGroup(147,1);`
`# by ID`

`G:=PCGroup([3,-3,-7,-7,541,46,380]);`
`// Polycyclic`

`G:=Group<a,b|a^49=b^3=1,b*a*b^-1=a^18>;`
`// generators/relations`

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