Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of #1113 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×C4×#1113 |
| Fitting | C34×C4 ⋊ C22 |
| derived | C33 ⋊ C22×C3×C4 |
| Sylow | C34 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | #1113 |
| commutator, hypocenter | C33 | C22×C3×C4 |
| Frattini | C2 | C2×C3×#1113 |
| Fitting | C34×C4 | C22 |
| socle | C2×C34 | C23 |
| prime radical | C2×C3×#1113 | C2 |
| cosocle | C2×C33 | C23×C3 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C3×C4 | |
774 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 774 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
180 conjugacy classes — view the full character table.
| maximal subgroup of | C4×(((C3⋊D3)⋊C9)⋊C2)⋊C3, C4×#11757730, C3×AGL(1,𝔽5)×#1113, … |
| maximal normal subgroup of | C4×#10620215, C3×D4×#1113 |
| maximal quotient of | C4×(#4207⋊C2)⋊C3, C3×AGL(1,𝔽5)×#1113 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3370);