Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #14618 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C23×D3 |
| commutator | C23×C3 | C23×C4 |
| Fitting | C2×C3×#399 | C2 |
| socle, cosocle | C24×C3 | C24 |
| prime radical | C25×C3 | C23 |
| hypercenter | C2×#399 | D3 |
| hypocenter | C3 | #14618 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C2×#399 | |
1290 conjugacy classes of subgroups; only the 327 normal subgroups are shown.
This group has 1290 conjugacy classes of subgroups — too many to draw. Only its 327 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 384 |
Polycyclic presentation:
2^8,3|a^2=h,b^2=efg,c^2=fh,d^2=efg,b^a=be,c^a=cf,d^a=def,i^a=i^2,c^b=cg,d^b=de,i^c=i^2108 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 100178);