Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8×C9 · Schur multiplier (H2) = C2×C32 |
| direct product | C2×C8×C32⋊C9 |
| unambiguous semidirect product | (C2×C32×C8)⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C8 | C32 |
| commutator | C3 | C2×C3×C8×C9 |
| Frattini, cosocle | C32×C4 | C22×C32 |
| socle | C22×C32 | C32×C4 |
| prime radical | C22×C33 | C3×C4 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C8 ◁ G | |
253 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 253 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2)(11,12,14)(13,16,18)(15,19,17)(20,21,23,22,24,26,25,27,28);(3,4,5,6,7,8,9,10)(12,15,16)(14,18,19)528 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 586);