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) = C37 |
| direct product | C3×C5×C9×H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center | C32×C5×C9 | C32 |
| commutator | C3 | C33×C5×C9 |
| Frattini, cosocle | C32 | C34×C5 |
| socle | C33×C5 | C33 |
| prime radical | C32×C5×H(3,𝔽3) | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C5×C9 ◁ G | |
1688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 1215 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2,3)(22,23,24,25,26);(4,5,6,7,8,9,10,11,12);(13,14,16)(15,18,20)(17,21,19);(14,17,18)(16,20,21)1485 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3645, 419);