Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #36676⋊C3 |
| Fitting, Sylow | #36676 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #560⋊C3 |
| commutator, cosocle, hypocenter | #95 | C23×C3 |
| Frattini, socle | C23 | C23×A4 |
| Fitting | #36676 | C3 |
| LCS | G ▷ #95 | |
| UCS | C1 ◁ C4 | |
451 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 451 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
3,2^8|b^2=g,f^2=h,b^a=bcdefg,c^a=ci,d^a=bci,e^a=egi,g^a=gi,i^a=g,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 | #72227331 |
| maximal normal subgroup of | #72227331 |
| maximal quotient of | #72222814, #72222820 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084425);