Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #14657⋊C3 |
| Fitting, Sylow | #14657 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #5450 |
| commutator | C25 | C2×C3×C4 |
| Frattini, socle | C24 | C22×A4 |
| Fitting | #14657 | C3 |
| prime radical | C22×#120⋊C3 | C2 |
| cosocle | C26 | C22×C3 |
| hypocenter | C24 | C3×C22⋊C4 |
| LCS | G ▷ C25 ▷ C24 | |
| UCS | C1 ◁ C22 | |
639 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 639 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
3,2^8|c^2=i,b^a=eghi,d^a=dfg,e^a=befgh,f^a=fg,g^a=f,c^b=cf,d^b=dg,d^c=dh,e^c=efg,e^d=ef44 conjugacy classes — view the full character table.
| maximal subgroup of | #72256365 |
| maximal normal subgroup of | #72256365 |
| maximal quotient of | #72256217 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083643);