Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #14873⋊C3 |
| Fitting, Sylow | #14873 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×#120⋊C3 |
| commutator, cosocle | C23×Q8 | C22×C3 |
| Frattini, socle | C24 | C22×A4 |
| Fitting | #14873 | C3 |
| hypercenter | C23 | #120⋊C3 |
| hypocenter | C22×Q8 | C3×D4 |
| LCS | G ▷ C23×Q8 ▷ C22×Q8 | |
| UCS | C1 ◁ C22 ◁ C23 | |
835 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 835 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
3,2^8|c^2=h,d^2=h,b^a=bg,c^a=degh,d^a=cdehi,e^a=eg,f^a=gi,g^a=fgi,c^b=cf,d^b=dg,e^b=ei,d^c=dh48 conjugacy classes — view the full character table.
| maximal subgroup of | #72227791, #72227796, #72227974, … |
| maximal normal subgroup of | #72227796, #72227985, #72267156, … |
| maximal quotient of | #72233782, #72267932, #72268816, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083665);