Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #36674⋊C3 |
| Fitting, Sylow | #36674 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | C22⋊A4 |
| commutator, cosocle, hypocenter | #2840 | C2×C3 |
| Frattini, socle | C23 | C2×C22⋊A4 |
| Fitting | #36674 | C3 |
| prime radical | #73293 | C2 |
| LCS | G ▷ #2840 | |
| UCS | C1 ◁ C22×C4 | |
443 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 443 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
3,2^8|b^2=h,d^2=h,f^2=g,b^a=bdgi,c^a=e,d^a=bgi,e^a=cegi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^d=egi60 conjugacy classes — view the full character table.
| maximal subgroup of | #72209974 |
| maximal normal subgroup of | #72209974 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084420);