Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting, Sylow | Q8×H(5,𝔽2) ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×C22⋊A4 |
| commutator, cosocle, hypocenter | Q82 | C22×C3 |
| Fitting | Q8×H(5,𝔽2) | C3 |
| hypercenter | C2×Q8 | C22⋊A4 |
| LCS | G ▷ Q82 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
1341 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 1341 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
3,2^8|b^2=h,c^2=h,b^a=cdi,c^a=bcghi,d^a=defghi,e^a=df,f^a=dei,g^a=dhi,c^b=ch,d^b=di,g^d=gi,f^e=fi55 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085167);