Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C23 |
| direct product | C4×#5935 |
| Fitting | C4×H(5,𝔽2) ⋊ D3 |
| derived | #997 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #1020 |
| commutator, hypocenter | #997 | C2×C4 |
| Frattini, socle | C22 | C2×#1020 |
| Fitting | C4×H(5,𝔽2) | D3 |
| prime radical | C2×#5935 | C2 |
| cosocle | C2×#997 | C22 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2×C4 | |
612 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 612 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 32 |
52 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×#5935, C2×C4×#5935, C4×SO+(4,𝔽3), … |
| maximal normal subgroup of | C2×C4×#5935, #72226121, D4×#5935, … |
| maximal quotient of | AGL(1,𝔽5)×#5935, C2×C4×#5935, C3×C4×#5935, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090058);