Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C26 |
| direct product | C23×#997 |
| Fitting, Sylow | C23×H(5,𝔽2) ⋊ C3 |
| derived | H(5,𝔽2) ⋊ C23×C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C24 | C22⋊A4 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C23×C3 |
| Frattini | C2 | C23×C22⋊A4 |
| Fitting | C23×H(5,𝔽2) | C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C24 | |
11481 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 11481 conjugacy classes of subgroups — too many to draw. Only its 179 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
88 conjugacy classes — view the full character table.
| maximal subgroup of | C23×Ω+(4,𝔽3), C23×#5935, #72226313, … |
| maximal normal subgroup of | C23×Ω+(4,𝔽3), C23×#5935, #72226313, … |
| maximal quotient of | C22×#35346⋊C3, C23×C22⋊#997, C2×D4×#997, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085302);