Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #33272⋊C3 |
| Fitting, Sylow | #33272 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C4⋊(#126⋊C3) |
| commutator, cosocle | #376 | C22×C3 |
| Frattini | C22×C4 | C22×A4 |
| Fitting | #33272 | C3 |
| socle | C23 | D4×A4 |
| hypercenter | C8 | #126⋊C3 |
| hypocenter | #95 | C3×D4 |
| LCS | G ▷ #376 ▷ #95 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
165 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 165 conjugacy classes of subgroups — too many to draw. Only its 23 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|c^2=fh,d^2=gi,e^2=f,g^2=i,b^a=bf,c^a=ceghi,d^a=dfhi,e^a=cgi,f^a=fhi,h^a=fi,c^b=cf,d^b=dg,e^b=eh,g^b=gi,e^c=ei36 conjugacy classes — view the full character table.
| maximal subgroup of | #72340609, #72340720, #72340730 |
| maximal normal subgroup of | #72340609, #72340730 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084023);