Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | #46324⋊C3 |
| Fitting, Sylow | #46324 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#560⋊C3 |
| commutator, cosocle | C2×C42 | C23×C3 |
| Frattini, socle | C23 | C23×A4 |
| Fitting | #46324 | C3 |
| hypercenter | C22 | #560⋊C3 |
| hypocenter | C42 | C2×C3×D4 |
| LCS | G ▷ C2×C42 ▷ C42 | |
| UCS | C1 ◁ C2 ◁ C22 | |
706 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 706 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
3,2^8|d^2=i,e^2=gi,f^2=i,b^a=bgi,c^a=cegi,d^a=dei,e^a=f,f^a=efg,g^a=gi,h^a=hi,i^a=g,c^b=cg,d^b=dh,e^b=ei,f^b=fg,d^c=di,e^c=egi,f^c=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72214043, #72219748, #72219750, … |
| maximal normal subgroup of | #72214043, #72219748, #72219750, … |
| maximal quotient of | #72213963, #72213984, #72214004, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084455);