metacyclic, supersoluble, monomial, Z-group, 7-hyperelementary
Aliases: C43⋊C7, SmallGroup(301,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C43 — C43⋊C7 |
Generators and relations for C43⋊C7
G = < a,b | a43=b7=1, bab-1=a21 >
Character table of C43⋊C7
| class | 1 | 7A | 7B | 7C | 7D | 7E | 7F | 43A | 43B | 43C | 43D | 43E | 43F | |
| size | 1 | 43 | 43 | 43 | 43 | 43 | 43 | 7 | 7 | 7 | 7 | 7 | 7 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | ζ75 | ζ74 | ζ76 | ζ7 | ζ73 | ζ72 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 7 |
| ρ3 | 1 | ζ76 | ζ72 | ζ73 | ζ74 | ζ75 | ζ7 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 7 |
| ρ4 | 1 | ζ7 | ζ75 | ζ74 | ζ73 | ζ72 | ζ76 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 7 |
| ρ5 | 1 | ζ73 | ζ7 | ζ75 | ζ72 | ζ76 | ζ74 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 7 |
| ρ6 | 1 | ζ72 | ζ73 | ζ7 | ζ76 | ζ74 | ζ75 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 7 |
| ρ7 | 1 | ζ74 | ζ76 | ζ72 | ζ75 | ζ7 | ζ73 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 7 |
| ρ8 | 7 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4334+ζ4330+ζ4329+ζ4328+ζ4326+ζ4318+ζ437 | ζ4336+ζ4325+ζ4317+ζ4315+ζ4314+ζ4313+ζ439 | ζ4341+ζ4335+ζ4321+ζ4316+ζ4311+ζ434+ζ43 | ζ4342+ζ4339+ζ4332+ζ4327+ζ4322+ζ438+ζ432 | ζ4340+ζ4338+ζ4331+ζ4324+ζ4323+ζ4310+ζ436 | ζ4337+ζ4333+ζ4320+ζ4319+ζ4312+ζ435+ζ433 | complex faithful |
| ρ9 | 7 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4337+ζ4333+ζ4320+ζ4319+ζ4312+ζ435+ζ433 | ζ4340+ζ4338+ζ4331+ζ4324+ζ4323+ζ4310+ζ436 | ζ4336+ζ4325+ζ4317+ζ4315+ζ4314+ζ4313+ζ439 | ζ4334+ζ4330+ζ4329+ζ4328+ζ4326+ζ4318+ζ437 | ζ4341+ζ4335+ζ4321+ζ4316+ζ4311+ζ434+ζ43 | ζ4342+ζ4339+ζ4332+ζ4327+ζ4322+ζ438+ζ432 | complex faithful |
| ρ10 | 7 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4336+ζ4325+ζ4317+ζ4315+ζ4314+ζ4313+ζ439 | ζ4334+ζ4330+ζ4329+ζ4328+ζ4326+ζ4318+ζ437 | ζ4342+ζ4339+ζ4332+ζ4327+ζ4322+ζ438+ζ432 | ζ4341+ζ4335+ζ4321+ζ4316+ζ4311+ζ434+ζ43 | ζ4337+ζ4333+ζ4320+ζ4319+ζ4312+ζ435+ζ433 | ζ4340+ζ4338+ζ4331+ζ4324+ζ4323+ζ4310+ζ436 | complex faithful |
| ρ11 | 7 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4342+ζ4339+ζ4332+ζ4327+ζ4322+ζ438+ζ432 | ζ4341+ζ4335+ζ4321+ζ4316+ζ4311+ζ434+ζ43 | ζ4340+ζ4338+ζ4331+ζ4324+ζ4323+ζ4310+ζ436 | ζ4337+ζ4333+ζ4320+ζ4319+ζ4312+ζ435+ζ433 | ζ4336+ζ4325+ζ4317+ζ4315+ζ4314+ζ4313+ζ439 | ζ4334+ζ4330+ζ4329+ζ4328+ζ4326+ζ4318+ζ437 | complex faithful |
| ρ12 | 7 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4341+ζ4335+ζ4321+ζ4316+ζ4311+ζ434+ζ43 | ζ4342+ζ4339+ζ4332+ζ4327+ζ4322+ζ438+ζ432 | ζ4337+ζ4333+ζ4320+ζ4319+ζ4312+ζ435+ζ433 | ζ4340+ζ4338+ζ4331+ζ4324+ζ4323+ζ4310+ζ436 | ζ4334+ζ4330+ζ4329+ζ4328+ζ4326+ζ4318+ζ437 | ζ4336+ζ4325+ζ4317+ζ4315+ζ4314+ζ4313+ζ439 | complex faithful |
| ρ13 | 7 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4340+ζ4338+ζ4331+ζ4324+ζ4323+ζ4310+ζ436 | ζ4337+ζ4333+ζ4320+ζ4319+ζ4312+ζ435+ζ433 | ζ4334+ζ4330+ζ4329+ζ4328+ζ4326+ζ4318+ζ437 | ζ4336+ζ4325+ζ4317+ζ4315+ζ4314+ζ4313+ζ439 | ζ4342+ζ4339+ζ4332+ζ4327+ζ4322+ζ438+ζ432 | ζ4341+ζ4335+ζ4321+ζ4316+ζ4311+ζ434+ζ43 | complex faithful |
►On 43 points: primitive
Generators in S43
(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)
(2 42 5 36 17 12 22)(3 40 9 28 33 23 43)(4 38 13 20 6 34 21)(7 32 25 39 11 24 41)(8 30 29 31 27 35 19)(10 26 37 15 16 14 18)
G:=sub<Sym(43)| (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), (2,42,5,36,17,12,22)(3,40,9,28,33,23,43)(4,38,13,20,6,34,21)(7,32,25,39,11,24,41)(8,30,29,31,27,35,19)(10,26,37,15,16,14,18)>;
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), (2,42,5,36,17,12,22)(3,40,9,28,33,23,43)(4,38,13,20,6,34,21)(7,32,25,39,11,24,41)(8,30,29,31,27,35,19)(10,26,37,15,16,14,18) );
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)], [(2,42,5,36,17,12,22),(3,40,9,28,33,23,43),(4,38,13,20,6,34,21),(7,32,25,39,11,24,41),(8,30,29,31,27,35,19),(10,26,37,15,16,14,18)]])
Matrix representation of C43⋊C7 ►in GL7(𝔽3011)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 195 | 1916 | 1073 | 194 | 1917 | 1072 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2771 | 3002 | 2479 | 372 | 273 | 2796 | 1647 |
| 2816 | 1095 | 1938 | 2817 | 1094 | 1939 | 1 |
| 1626 | 1636 | 1957 | 1668 | 2193 | 2992 | 2771 |
| 2988 | 2194 | 1282 | 2792 | 2390 | 1282 | 2816 |
| 721 | 2926 | 339 | 2904 | 2335 | 57 | 1626 |
| 657 | 185 | 372 | 830 | 208 | 373 | 2988 |
G:=sub<GL(7,GF(3011))| [0,0,0,0,0,0,1,1,0,0,0,0,0,195,0,1,0,0,0,0,1916,0,0,1,0,0,0,1073,0,0,0,1,0,0,194,0,0,0,0,1,0,1917,0,0,0,0,0,1,1072],[1,2771,2816,1626,2988,721,657,0,3002,1095,1636,2194,2926,185,0,2479,1938,1957,1282,339,372,0,372,2817,1668,2792,2904,830,0,273,1094,2193,2390,2335,208,0,2796,1939,2992,1282,57,373,0,1647,1,2771,2816,1626,2988] >;
C43⋊C7 in GAP, Magma, Sage, TeX
C_{43}\rtimes C_7 % in TeX
G:=Group("C43:C7"); // GroupNames label
G:=SmallGroup(301,1);
// by ID
G=gap.SmallGroup(301,1);
# by ID
G:=PCGroup([2,-7,-43,1149]);
// Polycyclic
G:=Group<a,b|a^43=b^7=1,b*a*b^-1=a^21>;
// generators/relations
Export
Subgroup lattice of C43⋊C7 in TeX
Character table of C43⋊C7 in TeX