Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | C4⋊(#393⋊C3) |
| unambiguous semidirect product | #15157⋊C3 |
| Fitting, Sylow | #15157 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×#120⋊C3 |
| commutator, cosocle | C23×Q8 | C22×C3 |
| Frattini, socle | C24 | C22×A4 |
| Fitting | #15157 | C3 |
| hypercenter | C2×C4 | #120⋊C3 |
| hypocenter | C22×Q8 | C3×D4 |
| LCS | G ▷ C23×Q8 ▷ C22×Q8 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
471 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 471 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
3,2^8|c^2=h,d^2=h,e^2=i,b^a=bg,c^a=dfgi,d^a=cdg,e^a=eg,f^a=g,g^a=fg,c^b=cf,d^b=dg,e^b=ei,d^c=dh48 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083677);