Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting, Sylow | C22×#577 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×C22⋊A4 |
| commutator, cosocle | C22×#41 | C22×C3 |
| Frattini | C4 | C22×C22⋊A4 |
| Fitting | C22×#577 | C3 |
| socle | C23 | D4×A4 |
| hypercenter | SD16 | C22⋊A4 |
| hypocenter | C22×Q8 | C3×D4 |
| LCS | G ▷ C22×#41 ▷ C22×Q8 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ SD16 | |
1953 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 1953 conjugacy classes of subgroups — too many to draw. Only its 50 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|b^2=i,h^2=i,b^a=behi,c^a=cefh,d^a=e,e^a=deh,f^a=gi,g^a=fgi,c^b=ch,h^b=hi,h^c=hi,e^d=ei48 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085270);