Finite Group DatabasebetaFinite Group Databasebeta| order | 26973 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9×C37 · Schur multiplier (H2) = C3 |
| direct product | C37×#10624520 |
| subgroup | G/H | |
|---|---|---|
| center | C9×C37 | #732 |
| commutator | C3×C9 | C3×C9×C37 |
| Frattini, cosocle | C32×C9 | C32×C37 |
| socle | C3×C37 | #6868 |
| prime radical | C33×C37 | 3−1+2 |
| LCS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C9×C37 ◁ C3×C9×C37 ◁ C32×C9×C37 ◁ G | |
124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 999 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,6,4,9,18,14,8,23,5,11,19,15,22,26,25,21,27,16,10,24,13,7,20,3,12,17)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64);(2,7,21,11,9,12,10,22,8)(6,17,20,24,27,26,19,23,18)3885 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26973, 80);