Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting, Sylow | #36575 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×#120⋊C3 |
| commutator, cosocle | C25 | C23×C3 |
| Frattini, socle | C23 | C23×A4 |
| Fitting | #36575 | C3 |
| hypercenter | Q8 | #120⋊C3 |
| hypocenter | C24 | C3×#41 |
| LCS | G ▷ C25 ▷ C24 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
994 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 994 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
3,2^8|b^2=i,c^2=i,d^2=i,e^2=i,f^2=i,c^a=cdfhi,d^a=cefg,e^a=eh,f^a=fg,g^a=gh,h^a=g,c^b=cg,d^b=dh,d^c=di,f^c=fi,e^d=ei,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72340611, #72340717, #72341520, … |
| maximal normal subgroup of | #72340611, #72341520, #41×#120⋊C3 |
| maximal quotient of | #72341582 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084382);