Finite Group DatabasebetaFinite Group Databasebeta| order | 38637 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C53 · Schur multiplier (H2) = C33 |
| direct product | C53×#10624769 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C53 | C3×H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C33×C53 |
| prime radical | C32×H(3,𝔽3)×C53 | C3 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C53 ◁ C34×C53 ◁ G | |
464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 4293 |
A minimal faithful permutation representation — 3 generators on 89 points:
(1,2,4,3,6,8,7,5,9)(10,11,14)(12,15,22)(13,16,23)(17,27,21)(18,24,30)(19,25,31)(20,26,32)(28,34,35)(29,33,36)(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)(4,8,9)(10,12,18)(11,15,24)(13,20,28)(14,22,30)(16,26,34)(17,25,33)(19,29,21)(23,32,35)(27,31,36);(11,16,17)(12,19,20)(18,28,29)(22,31,32)(24,33,34)(30,35,36)3021 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38637, 329);