Finite Group DatabasebetaFinite Group Databasebeta| order | 26973 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9×C37 · Schur multiplier (H2) = C9 |
| direct product | C37×#10624525 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C37 | #732 |
| commutator | C3×C9 | C3×C9×C37 |
| Frattini, cosocle | C32×C9 | C32×C37 |
| prime radical | C33×C37 | H(3,𝔽3) |
| LCS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C37 ◁ C33×C37 ◁ C32×C9×C37 ◁ G | |
154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 2997 |
A minimal faithful permutation representation — 2 generators on 73 points:
(1,2,4,3,5,7,6,8,9)(10,11,15)(12,19,14)(13,22,27)(16,28,31)(17,29,24)(18,30,25)(20,23,34)(21,33,26)(32,36,35)(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);(10,12,20,14,18,32,25,11,16)(13,17,30,24,21,15,26,22,19)(23,35,33,36,31,27,28,34,29)3885 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26973, 85);