Finite Group DatabasebetaFinite Group Databasebeta| order | 26973 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C35×C37 · Schur multiplier (H2) = C39 |
| direct product | C3×C37×3−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C37 | C34 |
| commutator, Frattini, cosocle | C3 | C35×C37 |
| prime radical | C32×H(3,𝔽3)×C37 | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C37 ◁ G | |
5588 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 2997 |
A minimal faithful permutation representation — 5 generators on 67 points:
(1,2,3)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67);(4,5,8,7,11,18,16,10,22)(6,12,23,15,21,17,14,9,19)(13,25,29,26,20,30,27,24,28);(4,6,13)(5,9,20)(7,15,26)(8,17,28)(10,21,25)(11,12,24)(14,27,16)(18,19,29)(22,23,30);(6,14,15)(9,21,12)(13,26,27)(17,23,19)(20,24,25)(28,29,30);(5,10,11)(8,18,22)(9,21,12)(17,19,23)(20,25,24)(28,29,30)9213 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26973, 502);