Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#87497 |
| Fitting | C2×#585 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C25 |
| Frattini | C2 | C24×S4 |
| Fitting | C2×#585 | D3 |
| socle | C22 | C23×S4 |
| hypercenter | C2×#41 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C4 ◁ C2×#41 | |
5305 conjugacy classes of subgroups; only the 511 normal subgroups are shown.
This group has 5305 conjugacy classes of subgroups — too many to draw. Only its 511 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
92 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72215767, C2×#11714992 |
| maximal normal subgroup of | C2×#11714992 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090171);