Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#87489 |
| Fitting | C23×#41 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×S4 |
| commutator, cosocle | C2×A4 | C25 |
| Frattini | C2 | C24×S4 |
| Fitting | C23×#41 | D3 |
| socle | C24 | C23×D3 |
| hypercenter | C2×#41 | S4 |
| hypocenter | A4 | C2×H(5,𝔽2) |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C22 ◁ C2×#41 | |
6049 conjugacy classes of subgroups; only the 517 normal subgroups are shown.
This group has 6049 conjugacy classes of subgroups — too many to draw. Only its 517 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
88 conjugacy classes — view the full character table.
| maximal subgroup of | #72259044, #72259082, C2×S4×H(5,𝔽2), … |
| maximal normal subgroup of | #72259044, #72259082, C2×S4×H(5,𝔽2) |
| maximal quotient of | #72259044, #72259082, C2×#72220618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090153);