Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×#32898⋊C3 |
| Fitting, Sylow | C2×#32898 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120⋊C3 |
| commutator, cosocle | C2×H(5,𝔽2) | C23×C3 |
| Frattini | C24 | C23×A4 |
| Fitting | C2×#32898 | C3 |
| hypercenter | C24 | #120⋊C3 |
| hypocenter | H(5,𝔽2) | C2×C3×D4 |
| LCS | G ▷ C2×H(5,𝔽2) ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C23 ◁ C24 | |
7236 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 7236 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
86 conjugacy classes — view the full character table.
| maximal subgroup of | #72268240, C2×#72227295, C2×#72218504, … |
| maximal normal subgroup of | C2×#72218504, C2×#72241382, C2×D4×B(4,𝔽2)⋊C3, … |
Past the SmallGroups library coverage — no SmallGroups id.