Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #36573⋊C3 |
| Fitting, Sylow | #36573 ⋊ C3 |
| derived | C42×Q8 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #3444⋊C3 |
| commutator, cosocle, hypocenter | C42×Q8 | C2×C3 |
| Frattini, socle | C23 | C2×C22⋊A4 |
| Fitting | #36573 | C3 |
| LCS | G ▷ C42×Q8 | |
| UCS | C1 ◁ C2 | |
392 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 24 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|c^2=ghi,d^2=ghi,e^2=hi,f^2=gi,b^a=bh,c^a=deg,d^a=cdfh,e^a=efh,f^a=ehi,g^a=h,h^a=gh,c^b=cg,d^b=dh,d^c=di,f^c=fi,e^d=ei,f^e=fi34 conjugacy classes — view the full character table.
| maximal subgroup of | #72260547 |
| maximal quotient of | #72260940 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084380);