Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | S4×H(5,𝔽2) |
| Fitting | C22×H(5,𝔽2) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C24×S4 |
| commutator, cosocle | C2×A4 | C25 |
| Fitting | C22×H(5,𝔽2) | D3 |
| socle | C23 | C24×D3 |
| hypercenter | H(5,𝔽2) | S4 |
| hypocenter | A4 | C2×H(5,𝔽2) |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2 ◁ H(5,𝔽2) | |
6067 conjugacy classes of subgroups; only the 513 normal subgroups are shown.
This group has 6067 conjugacy classes of subgroups — too many to draw. Only its 513 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
85 conjugacy classes — view the full character table.
| maximal subgroup of | S4×B(4,𝔽2), #72228314, #72241425, … |
| maximal normal subgroup of | S4×B(4,𝔽2), S4×#997, S4×#453, … |
| maximal quotient of | #72220683, H(5,𝔽2)×GL(2,𝔽3), #72228314, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090156);