Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting, Sylow | #33999 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×A4 |
| commutator, hypocenter | Q8 | C2×C3×C42 |
| Frattini, socle | C23 | C23×A4 |
| Fitting | #33999 | C3 |
| prime radical | C22×#41⋊C3 | C22 |
| cosocle | C22×Q8 | C23×C3 |
| hypercenter | #376 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C22×C4 ◁ #376 | |
838 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 838 conjugacy classes of subgroups — too many to draw. Only its 173 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=h,c^2=i,d^2=h,f^2=g,b^a=deghi,e^a=bdefi,c^b=cg,e^b=eg,d^c=dg152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084135);