Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
| direct product | C23×C4⋊S4 |
| Fitting | C25×C4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C24 | C2×S4 |
| commutator, cosocle | C2×A4 | C25 |
| Frattini | C2 | C24×S4 |
| Fitting | C25×C4 | D3 |
| socle | C26 | C2×D3 |
| hypercenter | C23×C4 | S4 |
| hypocenter | A4 | C23×D4 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C24 ◁ C23×C4 | |
9681 conjugacy classes of subgroups; only the 661 normal subgroups are shown.
This group has 9681 conjugacy classes of subgroups — too many to draw. Only its 661 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
112 conjugacy classes — view the full character table.
| maximal subgroup of | C23×D4×S4, #72259051, #72259162, … |
| maximal normal subgroup of | C23×D4×S4, #72259051, #72259162, … |
| maximal quotient of | C22×#85440, C22×#11713272, C2×D4×C4⋊S4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090143);