Finite Group DatabasebetaFinite Group Databasebeta| order | 26973 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C37 · Schur multiplier (H2) = C32 |
| direct product | C37×#10624607 |
| 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 | |
452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 2997 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,4,3,5,8,7,6,9)(11,13,15)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55);(2,5,6)(10,11,14,12,15,18,16,13,17);(2,6,5)(4,8,9)2701 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26973, 167);