Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33×C8 · Schur multiplier (H2) = C2×C32 |
| direct product | C2×C8×#737 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C9 | C32 |
| commutator | C3 | C2×C33×C8 |
| Frattini, cosocle | C3×C4 | C22×C33 |
| socle | C22×C3 | C33×C4 |
| prime radical | C22×H(3,𝔽3) | C3×C4 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C9 ◁ G | |
363 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2)(11,12,15)(13,17,23)(14,18,22)(16,24,21)(19,26,31)(20,27,30)(25,32,29)(28,34,36)(33,37,35);(3,4,5,6,7,8,9,10)(12,16,18)(15,22,24)(17,25,27)(23,30,32)(26,33,34)(31,36,37);(11,13,19,14,20,28,21,29,35)(12,17,26,18,27,34,16,25,33)(15,23,31,22,30,36,24,32,37)528 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1762);