Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#1401 ⋊ C2 |
| Sylow | C3 ⋊ #13669 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #69108 |
| commutator, cosocle | C2×C3×C16 | C23 |
| Frattini | C2×C16 | C22×D3 |
| Fitting | C3×#1401 | C2 |
| socle | C2×C3 | #2184 |
| hypercenter | #1401 | D3 |
| hypocenter | C3 | #13669 |
| LCS | G ▷ C2×C3×C16 ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #504 ◁ #1401 | |
288 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^8,3|d^2=fg,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fg,g^a=gh,i^a=i^2,d^b=df,f^b=fg,g^b=gh,i^b=i^2,e^c=eh93 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 90254);