Finite Group DatabasebetaFinite Group Databasebeta| order | 26973 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C9×C37 · Schur multiplier (H2) = C34 |
| direct product | C37×#10624689 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C37 | C3×H(3,𝔽3) |
| commutator | C32 | C32×C9×C37 |
| Frattini, cosocle | C33 | C33×C37 |
| prime radical | C32×H(3,𝔽3)×C37 | C3 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C37 ◁ C34×C37 ◁ G | |
512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 2997 |
A minimal faithful permutation representation — 3 generators on 73 points:
(1,2,4,3,5,7,6,8,9)(10,11,14)(12,18,23)(13,17,24)(15,25,21)(16,28,22)(19,30,33)(20,27,34)(26,36,32)(29,35,31)(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);(10,12,19)(11,15,26)(13,21,31)(14,23,33)(16,18,29)(17,27,30)(20,32,22)(24,25,35)(28,34,36);(12,20,21)(15,18,27)(19,31,32)(23,34,25)(26,30,29)(33,35,36)3885 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26973, 249);