Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C23×D3×S4 (order 1152) · Inn(G) = C22×D3 · Out(G) = C23×D3 |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×D3×#41 |
| Fitting | C32×#41 ⋊ C2 |
| Sylow | C32 ⋊ C2×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C22×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 | C2×C3×#41 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C3×C4 ◁ C3×#41 | |
348 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 174 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:
(1,7,4,3)(2,8,5,6)(9,10,11)(12,13,14);(2,5)(6,8);(1,2)(3,6)(4,5)(7,8);(13,14)90 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#837, #10591670, C3×D3×#104, … |
| maximal normal subgroup of | D3×#837, #10591670, C3×D3×#104, … |
| maximal quotient of | C3×#5517, C3×D3×#117, C3×#5522, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 998);