Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #32706⋊C3 |
| Fitting, Sylow | #32706 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #2812⋊C3 |
| commutator, cosocle | C24×C4 | C22×C3 |
| Frattini | C22×C4 | C22×A4 |
| Fitting | #32706 | C3 |
| socle | C23 | D4×A4 |
| hypercenter | C4 | #5450 |
| hypocenter | C24 | C3×D8 |
| LCS | G ▷ C24×C4 ▷ C25 ▷ C24 | |
| UCS | C1 ◁ C2 ◁ C4 | |
250 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 24 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|h^2=i,b^a=ei,c^a=cg,d^a=dfg,e^a=be,f^a=fg,g^a=f,c^b=cf,d^b=dg,d^c=dh,e^c=efg,h^c=hi,e^d=ef,h^d=hi32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083947);