Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×#85564 |
| derived | #997 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C23 | #1020 |
| commutator, hypocenter | #997 | C2×C4 |
| Frattini | C22 | C2×#1020 |
| Fitting | C22×H(5,𝔽2) | D3 |
| prime radical | C22×#997 | C2 |
| cosocle | C2×#997 | C22 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C23 | |
1255 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 1255 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
52 conjugacy classes — view the full character table.
| maximal subgroup of | #72227756, #72227773, #72227971, … |
| maximal normal subgroup of | #72227773, #72227984, C2×C4×#5934, … |
| maximal quotient of | C2×#14514394, #72246416, C2×#72210084, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090037);