Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#1010 |
| Fitting | C32×#41 ⋊ C2 |
| Sylow | C32 ⋊ 2−1+4 |
| 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 | C3×2−1+4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C3 ◁ C3×#41 | |
312 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 312 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) = 22 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2,3)(4,5,6)(7,10,12,11)(8,19,15,14)(9,18,17,16)(13,20,22,21);(7,11,12,10)(8,19,15,14)(9,18,17,16)(13,21,22,20);(7,17,12,9)(8,22,15,13)(10,16,11,18)(14,20,19,21);(5,6)(7,8)(9,13)(10,14)(11,19)(12,15)(16,20)(17,22)(18,21)81 conjugacy classes — view the full character table.
| maximal subgroup of | #10591684, #10591685, #10591687, … |
| maximal normal subgroup of | #10591684, #10591685, #10591687, … |
| maximal quotient of | C3×#5516, C3×#5518, C3×#5519, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 1000);