Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#1009 |
| Fitting | C32×#41 ⋊ C2 |
| Sylow | C32 ⋊ H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C23×D3 |
| commutator, cosocle | C2×C3 | C24×C3 |
| Frattini | C2 | C23×C3×D3 |
| Fitting | C32×#41 | C2 |
| socle | C2×C32 | C24 |
| hypercenter | C3×#41 | D3 |
| hypocenter | C3 | C3×H(5,𝔽2) |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C3 ◁ C3×#41 | |
352 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 352 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 14 points:
(1,2,3)(4,5,6)(7,13,10,9)(8,14,11,12);(8,11)(12,14);(7,8)(9,12)(10,11)(13,14);(5,6)(9,13)(12,14)81 conjugacy classes — view the full character table.
| maximal subgroup of | #10591664, #10591665, #10591666, … |
| maximal normal subgroup of | #10591664, #10591665, #10591666, … |
| maximal quotient of | C3×#5519, C3×#5521, C3×#5523, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 999);