Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting, derived, Sylow | #63048 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C22 | C22⋊(C22⋊A4) |
| commutator, Fitting, cosocle, hypocenter | #63048 | C3 |
| LCS | G ▷ #63048 | |
| UCS | C1 ◁ C22 | |
1297 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 1297 conjugacy classes of subgroups — too many to draw. Only its 50 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,c^2=h,f^2=i,g^2=i,b^a=c,c^a=bc,d^a=ei,e^a=deh,f^a=ghi,g^a=fghi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^c=ei,g^f=gi34 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085233);