Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| unambiguous semidirect product | #62893⋊C3 |
| Fitting, Sylow | #62893 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C24×A4 |
| commutator, cosocle | C2×Q8 | C24×C3 |
| Fitting | #62893 | C3 |
| hypercenter | #534 | A4 |
| hypocenter | Q8 | C3×H(5,𝔽2) |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ #534 | |
1593 conjugacy classes of subgroups; only the 217 normal subgroups are shown.
This group has 1593 conjugacy classes of subgroups — too many to draw. Only its 217 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
3,2^8|c^2=h,d^2=h,b^a=bchi,c^a=dghi,d^a=cdi,c^b=ch,d^b=di,e^b=eh,g^b=gi,d^c=dh,f^e=fi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72268063, #72268142, #72268160, … |
| maximal normal subgroup of | #72268142, #72268160, #72271322, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085135);