Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
| direct product | C22×#5932 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C23 | #1020 |
| commutator, cosocle, hypocenter | #997 | C23 |
| Frattini | C2 | C22×#1020 |
| Fitting | C22×H(5,𝔽2) | D3 |
| prime radical | C22×#997 | C2 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C23 | |
1709 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 1709 conjugacy classes of subgroups — too many to draw. Only its 52 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 | #72227713, #72227727, C2×D3×#5932, … |
| maximal normal subgroup of | #72227727, C22×#87519, C2×#11715841, … |
| maximal quotient of | C2×D3×#5932, C22×#85564, C2×#11715588, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090214);