Finite Group DatabasebetaFinite Group Databasebeta| order | 38637 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C53 · Schur multiplier (H2) = C34 |
| direct product | C3×C53×#6856 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C33×C53 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C33×C53 |
| LCS | G ▷ C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C33×C53 ◁ C34×C53 ◁ G | |
692 conjugacy classes of subgroups.
| minimal degree | μ(G) = 74 |
| transitive degree | μt(G) = 12879 |
A minimal faithful permutation representation — 3 generators on 74 points:
(1,2,3)(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74);(4,5,7)(6,8,11)(9,12,10)(14,16,18);(5,8,9)(13,14,17)(15,18,21)(16,20,19)5565 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38637, 138);