Finite Group DatabasebetaFinite Group Databasebeta| order | 3645 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #6881 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C5 · Schur multiplier (H2) = C33 |
| direct product | C3×C5×#6882 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C5 | #732 |
| commutator, Frattini, cosocle | C3×C9 | C33×C5 |
| prime radical | C33×C5 | H(3,𝔽3) |
| LCS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C5 ◁ C33×C5 ◁ C32×C5×C9 ◁ G | |
200 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);(4,5,8,7,12,18,16,11,24)(6,13,17,14,10,22,25,23,30)(9,19,26,21,28,27,20,29,15);(5,9,10,12,21,23,11,20,13)(6,14,25)(8,17,28,24,30,19,18,22,29)(15,27,26)285 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3645, 403);