Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C33 · Schur multiplier (H2) = C2×C35 |
| direct product | C2×C32×C3⋊((C3⋊D3)⋊C3) |
| Fitting | C2×C33×H(3,𝔽3) ⋊ C2 |
| derived | C33 ⋊ C22×C33 |
| Sylow | C33×H(3,𝔽3) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32 | C3⋊((C3⋊D3)⋊C3) |
| commutator, cosocle, hypocenter | C33 | C22×C33 |
| Frattini | C3 | C2×C33×C3⋊D3 |
| Fitting | C2×C33×H(3,𝔽3) | C2 |
| socle | C2×C34 | C3×D3 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C2×C32 | |
7136 conjugacy classes of subgroups; only the 468 normal subgroups are shown.
This group has 7136 conjugacy classes of subgroups — too many to draw. Only its 468 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 3 generators on 20 points:
(8,9)(11,16,18,13,14,17)(12,15);(4,5,6)(7,8,9)(11,14,18)(13,16,17)(19,20);(1,2,3)(10,11,13)(12,14,17)(15,18,16)378 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C3×#13091503, C2×C3×#13091647, C2×#72221909 |
| maximal quotient of | C2×#72221916, C2×#72221892, C2×C3×#13091503, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((8,9)(11,16,18,13,14,17)(12,15), (4,5,6)(7,8,9)(11,14,18)(13,16,17)(19,20), (1,2,3)(10,11,13)(12,14,17)(15,18,16));