Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#951 |
| Sylow | C32 ⋊ #137 |
140 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 140 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)(9,16)(10,18)(12,19)(14,15)(20,21);(7,9)(8,12)(10,17)(11,16)(13,21)(14,19)(15,20)(18,22);(5,6)(7,15,11,8)(9,20,16,12)(10,14,18,13)(17,19,22,21)63 conjugacy classes — view the full character table.
| maximal subgroup of | #10591664, #10591668, #10591680, … |
| maximal normal subgroup of | #10591664, #10591668, #10591680, … |
| maximal quotient of | C3×#4995, C3×#4996, C3×#5004, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 722);