Finite Group DatabasebetaFinite Group Databasebeta| order | 26973 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C37 · Schur multiplier (H2) = C32 |
| direct product | C37×#10624673 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C37 | C3×H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C33×C37 |
| prime radical | C34×C37 | C32 |
| LCS | G ▷ C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32×C37 ◁ C34×C37 ◁ G | |
236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 91 |
| transitive degree | μt(G) = 2997 |
A minimal faithful permutation representation — 3 generators on 91 points:
(1,2,5,13,7,19,4,8,15)(3,9,14,11,18,26,12,6,16)(10,21,24,23,17,27,22,20,25)(28,29,32,31,35,42,40,34,46)(30,36,47,39,33,43,38,45,41)(37,44,51,49,53,54,50,48,52)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91);(1,3,10)(2,6,17)(4,12,22)(5,14,24)(7,9,20)(8,18,21)(11,23,13)(15,16,25)(19,26,27)(28,30,37,31,39,49,40,38,50)(29,33,44,35,45,53,34,36,48)(32,41,51,42,47,54,46,43,52);(2,7,8)(3,11,12)(5,15,19)(6,18,9)(10,22,23)(24,27,25)(29,34,35)(32,42,46)(33,36,45)(41,47,43)(44,48,53)(51,54,52)1517 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26973, 233);