Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of #1113 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×C3×#1113 |
| Fitting, socle | C2×C34 ⋊ C22 |
| derived | C33 ⋊ C23×C3 |
| Sylow | C34 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #1113 |
| commutator, cosocle, hypocenter | C33 | C23×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C34 | C22 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C2×C3 | |
474 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 474 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
90 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#88932, C3×#88939, #13049438, … |
| maximal normal subgroup of | C3×#88932, C3×#88939, #13049438, … |
| maximal quotient of | C3×#89040, C3×#89041, C3×#89042, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 748);