Finite Group DatabasebetaFinite Group Databasebeta| order | 38637 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C34×C53 · Schur multiplier (H2) = C34 |
| direct product | C53×#10624883 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C53 | C34 |
| commutator, Frattini, cosocle | C32 | C34×C53 |
| prime radical | C32×H(3,𝔽3)×C53 | C3 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32×C53 ◁ G | |
926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 4293 |
A minimal faithful permutation representation — 4 generators on 89 points:
(1,2,4,3,6,8,7,5,9)(10,11,14)(12,18,29)(13,17,24)(15,25,20)(16,28,22)(19,31,35)(21,27,23)(26,36,32)(30,34,33)(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);(10,12,19)(11,15,26)(13,21,32)(14,23,34)(16,27,31)(17,18,30)(20,33,22)(24,25,35)(28,29,36);(2,5,6)(4,8,9)(12,20,21)(15,27,18)(19,32,33)(23,29,25)(26,30,31)(34,35,36);(11,16,17)(14,24,28)(15,27,18)(23,25,29)(26,31,30)(34,35,36)5565 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38637, 443);