Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of #10620215 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10620215 |
| Fitting, socle | C2×C34 ⋊ C23 |
| derived, Sylow | C34 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #10620215 |
| commutator, cosocle, hypocenter | C34 | C24 |
| Frattini | C1 | G |
| Fitting, socle | C2×C34 | C23 |
| LCS | G ▷ C34 | |
| UCS | C1 ◁ C2 | |
1722 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 1722 conjugacy classes of subgroups — too many to draw. Only its 171 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
84 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13049813, C2×((((C3⋊D3)⋊C3)⋊D3)⋊C3)⋊C2, #72219714, … |
| maximal normal subgroup of | C2×#13049813, #72219714, #72219716, … |
| maximal quotient of | #72219714, C2×#13049549, #72220250, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3585);