Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#1747 ⋊ C2 |
| Sylow | C3 ⋊ #23794 |
| subgroup | G/H | |
|---|---|---|
| center | C8 | #874 |
| commutator, cosocle | C22×C3×C4 | C24 |
| Frattini | C22×C4 | C23×D3 |
| Fitting | C3×#1747 | C2 |
| socle | C2×C3 | #2397 |
| hypercenter | #1747 | D3 |
| hypocenter | C3 | #23794 |
| LCS | G ▷ C22×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ #1747 | |
930 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 930 conjugacy classes of subgroups — too many to draw. Only its 191 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:
2^8,3|a^2=e,d^2=ef,e^2=h,b^a=be,c^a=cf,g^a=gh,c^b=cg,e^b=eh,f^b=fh,i^b=i^2,i^c=i^296 conjugacy classes — view the full character table.
| maximal subgroup of | #72270246 |
| maximal normal subgroup of | #72270246 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 200096);