Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C27 |
| unambiguous semidirect product | C22⋊#5465 |
| Fitting, derived, Sylow | C22×H(3,𝔽4) ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | C22⋊(C22⋊A4) |
| commutator, Fitting, cosocle, hypocenter | C22×H(3,𝔽4) | C3 |
| socle | C24 | C22⋊A4 |
| LCS | G ▷ C22×H(3,𝔽4) | |
| UCS | C1 ◁ C22 | |
1886 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 1886 conjugacy classes of subgroups — too many to draw. Only its 53 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=h,c^2=h,b^a=bcg,c^a=bfgi,d^a=defg,e^a=dfhi,f^a=g,g^a=fg,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^c=ei36 conjugacy classes — view the full character table.
| maximal subgroup of | #72209771, #72209836, #72210258, … |
| maximal normal subgroup of | #72209771, #72209836, #72210822, … |
| maximal quotient of | #72209768, #72213475, #72225756, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085029);