Finite Group DatabasebetaFinite Group Databasebeta| order | 33291 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C34×C137 · Schur multiplier (H2) = C35 |
| direct product | C137×3−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C137 | C34 |
| commutator, Frattini, cosocle | C3 | C34×C137 |
| prime radical | C3×H(3,𝔽3)×C137 | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C137 ◁ G | |
460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 164 |
| transitive degree | μt(G) = 3699 |
A minimal faithful permutation representation — 4 generators on 164 points:
(1,2,5,4,8,15,13,7,19)(3,9,20,12,18,14,11,6,16)(10,22,26,23,17,27,24,21,25)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164);(1,3,10)(2,6,17)(4,12,23)(5,14,25)(7,18,22)(8,9,21)(11,24,13)(15,16,26)(19,20,27);(3,11,12)(6,18,9)(10,23,24)(14,20,16)(17,21,22)(25,26,27);(2,7,8)(5,15,19)(6,18,9)(14,16,20)(17,22,21)(25,26,27)11371 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33291, 66);