Finite Group DatabasebetaFinite Group Databasebeta| order | 38637 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C34×C53 · Schur multiplier (H2) = C36 |
| direct product | C3×C53×#6893 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C33×C53 | C33 |
| commutator, Frattini, cosocle | C32 | C34×C53 |
| prime radical | C32×H(3,𝔽3)×C53 | C3 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C33×C53 ◁ G | |
1232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 92 |
| transitive degree | μt(G) = 12879 |
A minimal faithful permutation representation — 4 generators on 92 points:
(1,2,3)(40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92);(4,5,7)(6,9,11)(8,12,10)(13,14,17)(15,19,25)(16,20,24)(18,26,23)(21,28,33)(22,29,32)(27,34,31)(30,36,38)(35,39,37);(14,18,20)(17,24,26)(19,27,29)(25,32,34)(28,35,36)(33,38,39);(5,8,9)(7,11,12)(13,15,21,16,22,30,23,31,37)(14,19,28,20,29,36,18,27,35)(17,25,33,24,32,38,26,34,39)8109 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38637, 428);