Finite Group DatabasebetaFinite Group Databasebeta| order | 38637 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C27×C53 · Schur multiplier (H2) = C32 |
| direct product | C53×#10624444 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C9×C53 | H(3,𝔽3) |
| commutator | C32 | C3×C27×C53 |
| Frattini, cosocle | C32×C9 | C32×C53 |
| socle | C32×C53 | C32⋊C9 |
| prime radical | C34×C53 | C9 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C9×C53 ◁ C32×C9×C53 ◁ G | |
316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 4293 |
A minimal faithful permutation representation — 2 generators on 89 points:
(1,2,4)(3,5,8)(6,9,7)(10,11,14,12,15,20,17,22,27,13,16,21,18,23,28,25,30,33,19,24,29,26,31,34,32,35,36)(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);(2,5,6)8109 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38637, 4);