Finite Group DatabasebetaFinite Group Databasebeta| order | 3645 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C5×C9 · Schur multiplier (H2) = C34×C9 |
| direct product | C3×C5×#6887 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C5×C9 | C32 |
| commutator | C3 | C33×C5×C9 |
| Frattini, cosocle | C32 | C34×C5 |
| socle | C33×C5 | C33 |
| prime radical | C34×C5 | C32 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C5×C9 ◁ G | |
1052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 1215 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,2,3)(40,41,42,43,44);(4,5,7,6,8,10,9,11,12)(13,14,17)(15,19,25)(16,20,24)(18,26,23)(21,28,33)(22,29,32)(27,34,31)(30,36,38)(35,39,37);(14,18,20)(17,24,26)(19,27,29)(25,32,34)(28,35,36)(33,38,39);(13,15,21,16,22,30,23,31,37)(14,19,28,20,29,36,18,27,35)(17,25,33,24,32,38,26,34,39)1485 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3645, 418);