Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C215 |
| direct product | C2×#72210027 |
| Fitting, Sylow | C2×H(9,𝔽2) ⋊ C3 |
| derived | H(9,𝔽2) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22 | C22⋊(C22⋊(C22⋊A4)) |
| commutator, cosocle, hypocenter | H(9,𝔽2) | C2×C3 |
| Frattini | C2 | C2×C22⋊(C22⋊(C22⋊A4)) |
| Fitting | C2×H(9,𝔽2) | C3 |
| LCS | G ▷ H(9,𝔽2) | |
| UCS | C1 ◁ C22 | |
2774982 conjugacy classes of subgroups; only the 1063 normal subgroups are shown.
This group has 2774982 conjugacy classes of subgroups — too many to draw. Only its 1063 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
182 conjugacy classes — view the full character table.
| maximal subgroup of | #72256438, #72256538, C2×#72210031, … |
| maximal normal subgroup of | C2×#72210031, #72256770, C2×#72210040 |
Past the SmallGroups library coverage — no SmallGroups id.