Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #17308⋊C3 |
| Fitting, Sylow | #17308 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #5450 |
| commutator, cosocle | C23×Q8 | C22×C3 |
| Frattini, socle | C24 | C22×A4 |
| Fitting | #17308 | C3 |
| prime radical | C2×C22⋊SL(2,𝔽3) | C22 |
| hypocenter | C22×Q8 | C3×Q8 |
| LCS | G ▷ C23×Q8 ▷ C22×Q8 | |
| UCS | C1 ◁ C22 | |
229 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 229 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
3,2^8|b^2=fgh,c^2=fgi,d^2=h,e^2=fgh,b^a=bfg,c^a=bdh,d^a=cdefh,e^a=egi,f^a=g,g^a=fgi,h^a=ghi,c^b=cf,d^b=dg,e^b=ei,d^c=dh,e^c=efg,e^d=ef42 conjugacy classes — view the full character table.
| maximal subgroup of | #72279177 |
| maximal quotient of | #72299986 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083720);