Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C42 |
No aliases for this group.
| Fitting, derived, Sylow | #17309 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #5464 |
| commutator, Fitting, cosocle, hypocenter | #17309 | C3 |
| Frattini, socle | C24 | C22⋊A4 |
| LCS | G ▷ #17309 | |
| UCS | C1 ◁ C22 | |
306 conjugacy classes of subgroups.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
3,2^8|b^2=i,c^2=h,d^2=h,e^2=i,b^a=egi,c^a=cdh,d^a=chi,e^a=befh,f^a=g,g^a=fg,c^b=cf,d^b=dg,e^b=ei,d^c=dh,e^c=efg,e^d=ef26 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083721);