metacyclic, supersoluble, monomial, Z-group
Aliases: C49⋊C6, D49⋊C3, C7.F7, C49⋊C3⋊C2, SmallGroup(294,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C49 — C49⋊C6 |
Generators and relations for C49⋊C6
G = < a,b | a49=b6=1, bab-1=a31 >
Character table of C49⋊C6
| class | 1 | 2 | 3A | 3B | 6A | 6B | 7 | 49A | 49B | 49C | 49D | 49E | 49F | 49G | |
| size | 1 | 49 | 49 | 49 | 49 | 49 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ5 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ6 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ7 | 6 | 0 | 0 | 0 | 0 | 0 | 6 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F7 |
| ρ8 | 6 | 0 | 0 | 0 | 0 | 0 | -1 | ζ4948+ζ4931+ζ4930+ζ4919+ζ4918+ζ49 | ζ4947+ζ4938+ζ4936+ζ4913+ζ4911+ζ492 | ζ4943+ζ4939+ζ4933+ζ4916+ζ4910+ζ496 | ζ4937+ζ4932+ζ4929+ζ4920+ζ4917+ζ4912 | ζ4945+ζ4927+ζ4926+ζ4923+ζ4922+ζ494 | ζ4940+ζ4934+ζ4925+ζ4924+ζ4915+ζ499 | ζ4946+ζ4944+ζ4941+ζ498+ζ495+ζ493 | orthogonal faithful |
| ρ9 | 6 | 0 | 0 | 0 | 0 | 0 | -1 | ζ4945+ζ4927+ζ4926+ζ4923+ζ4922+ζ494 | ζ4946+ζ4944+ζ4941+ζ498+ζ495+ζ493 | ζ4940+ζ4934+ζ4925+ζ4924+ζ4915+ζ499 | ζ4948+ζ4931+ζ4930+ζ4919+ζ4918+ζ49 | ζ4943+ζ4939+ζ4933+ζ4916+ζ4910+ζ496 | ζ4947+ζ4938+ζ4936+ζ4913+ζ4911+ζ492 | ζ4937+ζ4932+ζ4929+ζ4920+ζ4917+ζ4912 | orthogonal faithful |
| ρ10 | 6 | 0 | 0 | 0 | 0 | 0 | -1 | ζ4943+ζ4939+ζ4933+ζ4916+ζ4910+ζ496 | ζ4937+ζ4932+ζ4929+ζ4920+ζ4917+ζ4912 | ζ4947+ζ4938+ζ4936+ζ4913+ζ4911+ζ492 | ζ4945+ζ4927+ζ4926+ζ4923+ζ4922+ζ494 | ζ4940+ζ4934+ζ4925+ζ4924+ζ4915+ζ499 | ζ4946+ζ4944+ζ4941+ζ498+ζ495+ζ493 | ζ4948+ζ4931+ζ4930+ζ4919+ζ4918+ζ49 | orthogonal faithful |
| ρ11 | 6 | 0 | 0 | 0 | 0 | 0 | -1 | ζ4937+ζ4932+ζ4929+ζ4920+ζ4917+ζ4912 | ζ4940+ζ4934+ζ4925+ζ4924+ζ4915+ζ499 | ζ4945+ζ4927+ζ4926+ζ4923+ζ4922+ζ494 | ζ4946+ζ4944+ζ4941+ζ498+ζ495+ζ493 | ζ4948+ζ4931+ζ4930+ζ4919+ζ4918+ζ49 | ζ4943+ζ4939+ζ4933+ζ4916+ζ4910+ζ496 | ζ4947+ζ4938+ζ4936+ζ4913+ζ4911+ζ492 | orthogonal faithful |
| ρ12 | 6 | 0 | 0 | 0 | 0 | 0 | -1 | ζ4946+ζ4944+ζ4941+ζ498+ζ495+ζ493 | ζ4943+ζ4939+ζ4933+ζ4916+ζ4910+ζ496 | ζ4948+ζ4931+ζ4930+ζ4919+ζ4918+ζ49 | ζ4947+ζ4938+ζ4936+ζ4913+ζ4911+ζ492 | ζ4937+ζ4932+ζ4929+ζ4920+ζ4917+ζ4912 | ζ4945+ζ4927+ζ4926+ζ4923+ζ4922+ζ494 | ζ4940+ζ4934+ζ4925+ζ4924+ζ4915+ζ499 | orthogonal faithful |
| ρ13 | 6 | 0 | 0 | 0 | 0 | 0 | -1 | ζ4940+ζ4934+ζ4925+ζ4924+ζ4915+ζ499 | ζ4948+ζ4931+ζ4930+ζ4919+ζ4918+ζ49 | ζ4946+ζ4944+ζ4941+ζ498+ζ495+ζ493 | ζ4943+ζ4939+ζ4933+ζ4916+ζ4910+ζ496 | ζ4947+ζ4938+ζ4936+ζ4913+ζ4911+ζ492 | ζ4937+ζ4932+ζ4929+ζ4920+ζ4917+ζ4912 | ζ4945+ζ4927+ζ4926+ζ4923+ζ4922+ζ494 | orthogonal faithful |
| ρ14 | 6 | 0 | 0 | 0 | 0 | 0 | -1 | ζ4947+ζ4938+ζ4936+ζ4913+ζ4911+ζ492 | ζ4945+ζ4927+ζ4926+ζ4923+ζ4922+ζ494 | ζ4937+ζ4932+ζ4929+ζ4920+ζ4917+ζ4912 | ζ4940+ζ4934+ζ4925+ζ4924+ζ4915+ζ499 | ζ4946+ζ4944+ζ4941+ζ498+ζ495+ζ493 | ζ4948+ζ4931+ζ4930+ζ4919+ζ4918+ζ49 | ζ4943+ζ4939+ζ4933+ζ4916+ζ4910+ζ496 | orthogonal 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 20 19 49 31 32)(3 39 37 48 12 14)(4 9 6 47 42 45)(5 28 24 46 23 27)(7 17 11 44 34 40)(8 36 29 43 15 22)(10 25 16 41 26 35)(13 33 21 38 18 30)
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), (2,20,19,49,31,32)(3,39,37,48,12,14)(4,9,6,47,42,45)(5,28,24,46,23,27)(7,17,11,44,34,40)(8,36,29,43,15,22)(10,25,16,41,26,35)(13,33,21,38,18,30)>;
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), (2,20,19,49,31,32)(3,39,37,48,12,14)(4,9,6,47,42,45)(5,28,24,46,23,27)(7,17,11,44,34,40)(8,36,29,43,15,22)(10,25,16,41,26,35)(13,33,21,38,18,30) );
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)], [(2,20,19,49,31,32),(3,39,37,48,12,14),(4,9,6,47,42,45),(5,28,24,46,23,27),(7,17,11,44,34,40),(8,36,29,43,15,22),(10,25,16,41,26,35),(13,33,21,38,18,30)]])
Matrix representation of C49⋊C6 ►in GL6(𝔽883)
| 650 | 876 | 836 | 306 | 797 | 814 |
| 69 | 719 | 62 | 22 | 375 | 866 |
| 17 | 86 | 736 | 79 | 39 | 392 |
| 491 | 508 | 577 | 344 | 570 | 530 |
| 353 | 844 | 861 | 47 | 697 | 40 |
| 843 | 313 | 804 | 821 | 7 | 657 |
| 391 | 836 | 85 | 492 | 821 | 413 |
| 329 | 804 | 391 | 782 | 344 | 476 |
| 445 | 577 | 101 | 430 | 22 | 492 |
| 475 | 62 | 453 | 15 | 147 | 554 |
| 132 | 539 | 868 | 460 | 47 | 438 |
| 470 | 861 | 423 | 555 | 79 | 408 |
G:=sub<GL(6,GF(883))| [650,69,17,491,353,843,876,719,86,508,844,313,836,62,736,577,861,804,306,22,79,344,47,821,797,375,39,570,697,7,814,866,392,530,40,657],[391,329,445,475,132,470,836,804,577,62,539,861,85,391,101,453,868,423,492,782,430,15,460,555,821,344,22,147,47,79,413,476,492,554,438,408] >;
C49⋊C6 in GAP, Magma, Sage, TeX
C_{49}\rtimes C_6 % in TeX
G:=Group("C49:C6"); // GroupNames label
G:=SmallGroup(294,1);
// by ID
G=gap.SmallGroup(294,1);
# by ID
G:=PCGroup([4,-2,-3,-7,-7,938,1410,514,4035,1351]);
// Polycyclic
G:=Group<a,b|a^49=b^6=1,b*a*b^-1=a^31>;
// generators/relations
Export
Subgroup lattice of C49⋊C6 in TeX
Character table of C49⋊C6 in TeX