Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #19083⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #19083⋊C3 |
| Fitting, Sylow | #19083 ⋊ C3 |
| derived | #564 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #564 | C22×C3 |
| Frattini | C42 | C22×A4 |
| Fitting | #19083 | C3 |
| socle | C22 | C2×#120⋊C3 |
| LCS | G ▷ #564 | |
| UCS | C1 | |
186 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
3,2^8|c^2=i,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=bfi,c^a=cdefgh,d^a=cfi,e^a=egi,f^a=fgi,g^a=f,h^a=i,i^a=hi,c^b=cf,d^b=dg,e^b=eh,f^b=fh,g^b=ghi,f^c=fh,g^c=gi,e^d=eh,f^d=fi,g^d=ghi24 conjugacy classes — view the full character table.
| maximal subgroup of | #72272596, #72272611 |
| maximal normal subgroup of | #72272596, #72272611 |
| maximal quotient of | #72340881 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083786);