Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #62979⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C25 |
| unambiguous semidirect product | #62979⋊C3 |
| Fitting, Sylow | #62979 ⋊ C3 |
| derived | #564 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #564 | C22×C3 |
| Frattini, socle | C22 | C22×C22⋊A4 |
| Fitting | #62979 | C3 |
| LCS | G ▷ #564 | |
| UCS | C1 | |
1085 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 1085 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
3,2^8|b^2=i,c^2=h,d^2=h,e^2=i,f^2=h,b^a=cdefghi,c^a=bfi,d^a=bcdfh,e^a=dghi,f^a=bcegh,g^a=cdfgi,h^a=hi,i^a=h,c^b=ch,d^b=di,e^b=ei,g^b=gh,e^c=ei,f^c=fh,g^c=gi,e^d=eh,f^d=fi33 conjugacy classes — view the full character table.
| maximal subgroup of | #72234974, #72272503, #72272595 |
| maximal normal subgroup of | #72272503, #72272595 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085160);