Finite Group DatabasebetaFinite Group Databasebeta| order | 26973 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C37 · Schur multiplier (H2) = C35 |
| direct product | C37×#10624761 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C37 | C32⋊H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C33×C37 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C37 ◁ C33×C37 ◁ G | |
476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 999 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,14,8,6,4,9,16)(3,10,20,12,19,15,13,7,17)(11,18,24,23,21,25,22,26,27)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64);(1,2,6)(3,7,17)(4,9,5)(8,16,14)(10,20,12)(11,18,25)(13,19,15)(21,27,23)(22,26,24);(1,3,11)(2,7,18)(4,13,22)(5,15,24)(6,17,25)(8,10,21)(9,19,26)(12,23,14)(16,20,27)2109 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26973, 321);