Finite Group DatabasebetaFinite Group Databasebeta| order | 26973 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C37 · Schur multiplier (H2) = C32 |
| direct product | C37×#10624494 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C37 | #732 |
| commutator, Frattini, cosocle | C32×C9 | C32×C37 |
| prime radical | C33×C37 | H(3,𝔽3) |
| LCS | G ▷ C32×C9 ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C37 ◁ C33×C37 ◁ C32×C9×C37 ◁ G | |
92 conjugacy classes of subgroups.
| minimal degree | μ(G) = 91 |
| transitive degree | μt(G) = 2997 |
A minimal faithful permutation representation — 2 generators on 91 points:
(1,2,5)(3,9,13)(4,8,14)(6,15,11)(7,18,12)(10,20,23)(16,26,27)(17,25,22)(19,24,21)(29,32,38,45,51,54,34,40,49)(33,41,42,44,52,50,43,48,39)(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);(1,3,10,4,11,21,12,22,27)(2,6,16,8,17,20,7,9,19)(5,13,23,14,15,24,18,25,26)(28,29,33)(30,32,39)(31,34,44)(35,38,48)(36,40,42)(37,45,43)(41,46,49)(47,51,50)(52,53,54)2109 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26973, 54);