Finite Group DatabasebetaFinite Group Databasebeta| order | 38637 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C9×C53 · Schur multiplier (H2) = C33 |
| direct product | C53×C27⋊H(3,𝔽3) |
| unambiguous semidirect product | C27⋊(H(3,𝔽3)×C53) |
| unambiguous semidirect product | (C27×C53)⋊H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center | C3×C9×C53 | C33 |
| commutator | C32 | C32×C9×C53 |
| Frattini, cosocle | C3×C9 | C33×C53 |
| socle | C32×C53 | C32×C9 |
| prime radical | C3×H(3,𝔽3)×C53 | C9 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C3×C9×C53 ◁ G | |
256 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 12879 |
A minimal faithful permutation representation — 3 generators on 89 points:
(1,2,4)(3,6,8)(5,9,7)(10,11,15,13,16,24,20,25,32,14,12,17,21,18,26,30,28,33,22,19,27,23,29,34,31,35,36)(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)(4,8,9);(10,12,17,13,18,26,20,28,33,14,19,27,21,29,34,30,35,36,22,11,15,23,16,24,31,25,32)8109 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38637, 275);