Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #11395⋊C3 |
| Fitting, Sylow | #11395 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #393⋊C3 |
| commutator, hypocenter | #543 | C3×C4 |
| Frattini | C23×C4 | C2×A4 |
| Fitting | #11395 | C3 |
| socle | C2 | #1446⋊C3 |
| prime radical | #543⋊C3 | C4 |
| cosocle | #3458 | C2×C3 |
| hypercenter | C2×C4 | #120⋊C3 |
| LCS | G ▷ #543 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
185 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 185 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
Polycyclic presentation:
3,2^8|b^2=h,c^2=g,d^2=g,h^2=i,b^a=bdefghi,c^a=defg,d^a=cdeh,e^a=fi,f^a=efgi,h^a=fgh,c^b=ce,d^b=df,e^b=ei,g^b=gi,d^c=dg,e^c=ei,h^c=hi,e^d=ei,f^d=fi42 conjugacy classes — view the full character table.
| maximal subgroup of | #72233415, #72233782, #72342896 |
| maximal normal subgroup of | #72233782, #72342896 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083557);