Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | C2×#73025 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22 | #1020⋊C2 |
| commutator, cosocle, hypocenter | #997 | C23 |
| Frattini | C23 | C22×S4 |
| Fitting | C2×B(4,𝔽2) | D3 |
| prime radical | C2×B(4,𝔽2)⋊C3 | C2 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C22 | |
588 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 588 conjugacy classes of subgroups — too many to draw. Only its 31 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 32 |
40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11714162, C2×#11714171, #72236902, … |
| maximal normal subgroup of | C2×#11714162, C2×#11714171, #72236902, … |
| maximal quotient of | #72245696, D3×#73025, C4×#73025, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088541);