Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting, Sylow | #62987 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C24×A4 |
| commutator, cosocle | C2×Q8 | C24×C3 |
| Fitting | #62987 | C3 |
| hypercenter | #548 | A4 |
| hypocenter | Q8 | C22×C3×D4 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ #548 | |
2371 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 2371 conjugacy classes of subgroups — too many to draw. Only its 239 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
3,2^8|b^2=i,b^a=bdgh,d^a=defgi,e^a=dfh,f^a=dehi,g^a=dhi,c^b=ch,d^b=di,g^d=gi,f^e=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72226051, #72226247, #72226333, … |
| maximal normal subgroup of | #72226051, #72226247, #72226333, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085176);