Finite Group DatabasebetaFinite Group Databasebeta| order | 26973 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C27×C37 · Schur multiplier (H2) = C34 |
| direct product | C27×H(3,𝔽3)×C37 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C27×C37 | C32 |
| commutator | C3 | C32×C27×C37 |
| Frattini, cosocle | C3×C9 | C33×C37 |
| socle | C32×C37 | C32×C9 |
| prime radical | C3×H(3,𝔽3)×C37 | C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C27×C37 ◁ G | |
292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 8991 |
A minimal faithful permutation representation — 3 generators on 73 points:
(10,11,14,12,15,20,17,22,27,13,16,21,18,23,28,25,30,33,19,24,29,26,31,34,32,35,36)(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);(1,2,4)(3,6,8)(5,9,7);(2,5,6)(4,8,9)10989 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26973, 274);