Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C25 |
| direct product | C22×B(4,𝔽2)⋊C3 |
| Fitting, Sylow | C22×B(4,𝔽2) ⋊ C3 |
| derived | H(5,𝔽2) ⋊ C23×C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C23 | #120⋊C3 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C23×C3 |
| Frattini | C23 | C23×A4 |
| Fitting | C22×B(4,𝔽2) | C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C23 | |
1454 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 1454 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 32 |
64 conjugacy classes — view the full character table.
| maximal subgroup of | #72227416, #72227705, #72227711, … |
| maximal normal subgroup of | #72227711, A4×B(4,𝔽2)⋊C3, C22×#73026, … |
| maximal quotient of | D4×B(4,𝔽2)⋊C3, C22×#2029⋊C3, C2×#32898⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084717);