metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C49⋊C3, C7.(C7⋊C3), SmallGroup(147,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C49 — C49⋊C3 |
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 |
►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
Export
Subgroup lattice of C49⋊C3 in TeX
Character table of C49⋊C3 in TeX