Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #213 |
| automorphism | Aut(G) = C23×D3×D4 (order 384) · Inn(G) = #67 · Out(G) = C24 |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#950 |
| Fitting | C32×#41 ⋊ C2 |
| Sylow | C32 ⋊ #135 |
144 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 58 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 — 3 generators on 22 points:
(1,2,3)(4,5,6)(8,13)(10,18)(12,19)(14,15)(16,22)(20,21);(7,17,11,9)(8,21,15,12)(10,22,18,16)(13,20,14,19);(5,6)(7,8)(9,12)(10,13)(11,15)(14,18)(16,19)(17,21)(20,22)72 conjugacy classes — view the full character table.
| maximal subgroup of | #10591673, #10591674, #10591675, … |
| maximal normal subgroup of | #10591673, #10591674, #10591675, … |
| maximal quotient of | C3×#4996, C3×#4998, C3×#5002, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 721);