Finite Group DatabasebetaFinite Group Databasebeta| order | 3645 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6875 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C5×C9 · Schur multiplier (H2) = C32 |
| direct product | C3×C5×#6875 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C5 | C9⋊C9 |
| commutator | C9 | C32×C5×C9 |
| Frattini, cosocle | C3×C9 | C33×C5 |
| prime radical | C33×C5 | 3−1+2 |
| LCS | G ▷ C9 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C5 ◁ C32×C5×C9 ◁ G | |
212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 405 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(31,32,33,34,35);(5,8,14,10,17,26,21,29,16)(6,11,22)(9,18,15,19,30,27,20,25,28)(12,24,23);(4,5,9,6,8,15,12,16,18,7,10,20,11,17,28,23,14,25,13,21,19,22,29,27,24,26,30)525 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3645, 268);