Finite Group DatabasebetaFinite Group Databasebeta| order | 26973 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C9×C37 · Schur multiplier (H2) = C33 |
| direct product | C9×C37×#733 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C9×C37 | H(3,𝔽3) |
| commutator | C32 | C32×C9×C37 |
| Frattini, cosocle | C33 | C33×C37 |
| socle | C32×C37 | C3×H(3,𝔽3) |
| prime radical | C33×C37 | C33 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C9×C37 ◁ C32×C9×C37 ◁ G | |
274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 8991 |
A minimal faithful permutation representation — 3 generators on 73 points:
(1,2,4,3,5,7,6,8,9)(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);(10,11,14,13,17,23,21,16,27)(12,18,22,20,15,24,31,26,34)(19,29,32,30,28,33,36,25,35);(10,12,19,21,31,36,13,20,30)(11,15,25,16,18,28,17,26,29)(14,22,32,27,34,35,23,24,33)5661 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26973, 279);