Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22×C4 |
| unambiguous semidirect product | #36644⋊C3 |
| Fitting, Sylow | #36644 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#127⋊C3 |
| commutator, cosocle | C2×C42 | C23×C3 |
| Frattini, socle | C23 | C23×A4 |
| Fitting | #36644 | C3 |
| hypercenter | C2×C4 | #127⋊C3 |
| hypocenter | C42 | C3×#41 |
| LCS | G ▷ C2×C42 ▷ C42 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
781 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 781 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
3,2^8|d^2=h,e^2=i,f^2=g,b^a=bgh,c^a=cgi,d^a=cdefgh,e^a=dgh,f^a=fgi,h^a=ghi,i^a=h,c^b=cg,d^b=dh,e^b=ei56 conjugacy classes — view the full character table.
| maximal subgroup of | #72225927, #72225958, #72225964, … |
| maximal normal subgroup of | #72225927, #72225958, #72225964, … |
| maximal quotient of | #72228375, #72258772, #72258777 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084410);