Finite Group DatabasebetaFinite Group Databasebeta| order | 26973 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C9×C37 · Schur multiplier (H2) = C35 |
| direct product | C3×C9×3−1+2×C37 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C9×C37 | C32 |
| commutator | C3 | C33×C9×C37 |
| Frattini, cosocle | C32 | C34×C37 |
| socle | C33×C37 | C33 |
| prime radical | C34×C37 | C32 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C9×C37 ◁ G | |
1148 conjugacy classes of subgroups; only the 872 normal subgroups are shown.
This group has 1148 conjugacy classes of subgroups — too many to draw. Only its 872 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 58 |
| transitive degree | μt(G) = 8991 |
A minimal faithful permutation representation — 4 generators on 58 points:
(1,2,3)(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58);(4,5,7,6,8,10,9,11,12);(13,14,16,15,18,20,19,17,21);(14,17,18)(16,20,21)10989 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26973, 420);