Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting, Sylow | #36592 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C23 | C23×A4 |
| commutator, cosocle | C22×Q8 | C23×C3 |
| Fitting | #36592 | C3 |
| hypercenter | #384 | A4 |
| hypocenter | Q8 | C3×#127 |
| LCS | G ▷ C22×Q8 ▷ Q8 | |
| UCS | C1 ◁ C23 ◁ #384 | |
932 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 932 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 256 |
Polycyclic presentation:
3,2^8|b^2=i,c^2=gi,d^2=hi,e^2=i,f^2=i,c^a=cefghi,d^a=degh,e^a=fgh,f^a=efghi,c^b=cg,d^b=dh,d^c=di,f^c=fi,e^d=ei,f^e=fi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084392);