Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #46333⋊C3 |
| Fitting, Sylow | #46333 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#560⋊C3 |
| commutator, cosocle | C2×C42 | C23×C3 |
| Frattini, socle | C23 | C23×A4 |
| Fitting | #46333 | C3 |
| hypercenter | C4 | #560⋊C3 |
| hypocenter | C42 | C3×#41 |
| LCS | G ▷ C2×C42 ▷ C42 | |
| UCS | C1 ◁ C2 ◁ C4 | |
414 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 414 conjugacy classes of subgroups — too many to draw. Only its 54 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|c^2=gh,d^2=ghi,e^2=gi,f^2=i,b^a=bgi,c^a=cefhi,d^a=defhi,e^a=ef,f^a=eghi,g^a=i,h^a=ghi,i^a=gi,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 | #72258819, #72258880, #72258934 |
| maximal normal subgroup of | #72258819, #72258880, #72258934 |
| maximal quotient of | #72258826 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084459);