Finite Group DatabasebetaFinite Group Databasebeta| order | 38637 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C53 · Schur multiplier (H2) = C32×C9 |
| direct product | C3×C53×#6881 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C53 | #732 |
| commutator, Frattini, cosocle | C3×C9 | C33×C53 |
| LCS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C53 ◁ C33×C53 ◁ C32×C9×C53 ◁ G | |
224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 4293 |
A minimal faithful permutation representation — 3 generators on 83 points:
(1,2,3)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83);(4,5,8)(6,13,17)(7,12,18)(9,19,15)(10,22,14)(11,24,16)(20,29,27)(21,28,26)(23,30,25);(5,9,10,12,21,23,11,20,13)(6,14,25)(8,17,28,24,30,19,18,22,29)(15,27,26)3021 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38637, 402);