Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C35 |
| direct product | C2×C3×C3⋊(C3⋊((C3⋊D3)⋊C3)) |
| Fitting | C2×C33×H(3,𝔽3) ⋊ C2 |
| derived | C34 ⋊ C22×C32 |
| Sylow | C33×H(3,𝔽3) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C3⋊(C3⋊((C3⋊D3)⋊C3)) |
| commutator, cosocle, hypocenter | C34 | C22×C32 |
| Frattini | C3 | C2×C32×C3⋊(C3⋊D3) |
| Fitting | C2×C33×H(3,𝔽3) | C2 |
| socle | C2×C34 | C3×D3 |
| LCS | G ▷ C34 | |
| UCS | C1 ◁ C2×C3 | |
7504 conjugacy classes of subgroups; only the 442 normal subgroups are shown.
This group has 7504 conjugacy classes of subgroups — too many to draw. Only its 442 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) = 324 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(8,9)(11,16,18,13,14,17)(12,15);(1,2,3)(7,8,9)(19,20);(10,11,13)(12,14,17)(15,18,16);(4,5,6)(10,11,13)(12,14,17)(15,18,16)324 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C3×#13091503 |
| maximal normal subgroup of | C2×C3×#13091503 |
| maximal quotient of | C2×#72221906, C2×#72221887, C2×#72221874, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,6)(8,9)(11,16,18,13,14,17)(12,15), (1,2,3)(7,8,9)(19,20), (10,11,13)(12,14,17)(15,18,16), (4,5,6)(10,11,13)(12,14,17)(15,18,16));