Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C22×C3×#67 |
| Fitting | C24×C32 ⋊ C2 |
| Sylow | C32 ⋊ C22×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C3 | C2×D3 |
| commutator, cosocle | C2×C3 | C24×C3 |
| Frattini | C2 | C23×C3×D3 |
| Fitting | C24×C32 | C2 |
| socle | C23×C32 | C22 |
| hypercenter | C24×C3 | D3 |
| hypocenter | C3 | C22×C3×D4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C23×C3 ◁ C24×C3 | |
539 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 539 conjugacy classes of subgroups — too many to draw. Only its 210 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:
(5,6,7)(8,9,10)(12,14);(6,7)(11,12)(13,14);(3,4);(1,2)108 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#63580, #10591707, #10591708, … |
| maximal normal subgroup of | C2×#63580, #10591707, #10591708, … |
| maximal quotient of | C22×C3×Dic3⋊C4, C2×C3×#924, C22×C3×(C2×D3)⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 1002);