Linear representation theory of Conway group:Co0

Smallest degree nontrivial irreducible representation
We know that the Conway group is the automorphism group of the Leech lattice. All automorphisms of a lattice are linear automorphisms, hence this action defines a linear representation of the Conway group $$\operatorname{Co}_0$$ on a 24-dimensional representation over the real numbers. This representation is irreducible over the complex numbers and is the smallest degree nontrivial irreducible representation.

Note that the representation is faithful, and it does not descend to an irreducible representation of Conway group:Co1.

Some information on its irreducible representations is, and we use the symbol "2.Co1" to access this information. The degrees of irreducible representations can be computed using the CharacterDegrees and CharacterTable functions.

gap> CharacterDegrees(CharacterTable("2.Co1")); [ [ 1, 1 ], [ 24, 1 ], [ 276, 1 ], [ 299, 1 ], [ 1771, 1 ], [ 2024, 1 ], [ 2576, 1 ], [ 4576, 1 ], [ 8855, 1 ], [ 17250, 1 ], [ 27300, 1 ],  [ 37674, 1 ], [ 40480, 1 ], [ 44275, 1 ], [ 80730, 1 ], [ 94875, 1 ],  [ 95680, 1 ], [ 170016, 1 ], [ 299000, 1 ], [ 313950, 1 ], [ 315744, 1 ],  [ 345345, 1 ], [ 351624, 1 ], [ 376740, 1 ], [ 388080, 1 ], [ 483000, 1 ],  [ 644644, 1 ], [ 673750, 2 ], [ 789360, 1 ], [ 822250, 1 ], [ 871884, 1 ],  [ 1434510, 1 ], [ 1450449, 1 ], [ 1771000, 1 ], [ 1821600, 1 ],  [ 1841840, 1 ], [ 1937520, 1 ], [ 2055625, 1 ], [ 2417415, 1 ],  [ 2464749, 2 ], [ 2816856, 1 ], [ 2877875, 1 ], [ 4004000, 1 ],  [ 4100096, 1 ], [ 5051904, 1 ], [ 5494125, 1 ], [ 5801796, 1 ],  [ 6446440, 2 ], [ 7104240, 1 ], [ 7628985, 1 ], [ 9152000, 2 ],  [ 9221850, 1 ], [ 9669660, 1 ], [ 11051040, 1 ], [ 12432420, 1 ],  [ 13156000, 1 ], [ 15002624, 2 ], [ 15471456, 1 ], [ 16170000, 2 ],  [ 16347825, 1 ], [ 17050176, 1 ], [ 17310720, 1 ], [ 18987696, 1 ],  [ 19734000, 1 ], [ 20083140, 1 ], [ 21049875, 1 ], [ 21528000, 1 ],  [ 21579129, 1 ], [ 23244375, 1 ], [ 24174150, 1 ], [ 24667500, 1 ],  [ 24794000, 1 ], [ 25900875, 1 ], [ 31574400, 1 ], [ 34155000, 1 ],  [ 40166280, 1 ], [ 40370176, 3 ], [ 44013375, 1 ], [ 44204160, 1 ],  [ 46621575, 1 ], [ 49335000, 1 ], [ 50519040, 1 ], [ 51571520, 2 ],  [ 55255200, 1 ], [ 57544344, 1 ], [ 59153976, 3 ], [ 60435375, 1 ],  [ 62790000, 1 ], [ 65270205, 1 ], [ 66602250, 1 ], [ 67358720, 1 ],  [ 77702625, 1 ], [ 83720000, 1 ], [ 85250880, 1 ], [ 91547820, 1 ],  [ 100725625, 1 ], [ 106142400, 1 ], [ 106260000, 1 ], [ 109882500, 1 ],  [ 112519680, 1 ], [ 139243104, 1 ], [ 150732800, 1 ], [ 161161000, 1 ],  [ 163478250, 1 ], [ 184184000, 1 ], [ 185912496, 1 ], [ 185955000, 1 ],  [ 190417920, 1 ], [ 191102976, 1 ], [ 201451250, 1 ], [ 205395750, 1 ],  [ 207491625, 1 ], [ 210496000, 1 ], [ 210974400, 1 ], [ 215547904, 2 ],  [ 219648000, 1 ], [ 230230000, 1 ], [ 241741500, 1 ], [ 247235625, 1 ],  [ 247543296, 1 ], [ 251756505, 1 ], [ 257857600, 1 ], [ 259008750, 1 ],  [ 267014475, 1 ], [ 280280000, 1 ], [ 282906624, 1 ], [ 287006720, 1 ],  [ 292953024, 1 ], [ 299710125, 1 ], [ 302176875, 1 ], [ 309429120, 1 ],  [ 313524224, 1 ], [ 326956500, 1 ], [ 342752256, 1 ], [ 351624000, 1 ],  [ 360062976, 1 ], [ 387317700, 1 ], [ 394680000, 1 ], [ 402902500, 1 ],  [ 464143680, 1 ], [ 464257024, 1 ], [ 469945476, 2 ], [ 483483000, 1 ],  [ 485760000, 1 ], [ 502078500, 1 ], [ 503513010, 1 ], [ 504627200, 1 ],  [ 517899096, 1 ], [ 522161640, 1 ], [ 551675124, 1 ], [ 557865000, 1 ],  [ 655360000, 1 ], [ 805805000, 1 ], [ 1021620600, 1 ] ]