Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×#73167 |
| Fitting, Sylow | C2×#2276 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×A4 |
| commutator, hypocenter | Q8 | C2×C3×C42 |
| Frattini | C23 | C23×A4 |
| Fitting | C2×#2276 | C3 |
| prime radical | C23×SL(2,𝔽3) | C22 |
| cosocle | C22×Q8 | C23×C3 |
| hypercenter | C2×#95 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C24 ◁ C2×#95 | |
974 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 974 conjugacy classes of subgroups — too many to draw. Only its 213 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 256 |
152 conjugacy classes — view the full character table.
| maximal subgroup of | #72268680 |
| maximal normal subgroup of | #72268680 |
| maximal quotient of | C2×#72222564 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084132);