Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #14885⋊C3 |
| Fitting, Sylow | #14885 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×#120⋊C3 |
| commutator, cosocle | C23×Q8 | C22×C3 |
| Frattini, socle | C24 | C22×A4 |
| Fitting | #14885 | C3 |
| hypercenter | C2×C4 | #120⋊C3 |
| hypocenter | C22×Q8 | C3×D4 |
| LCS | G ▷ C23×Q8 ▷ C22×Q8 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
881 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 881 conjugacy classes of subgroups — too many to draw. Only its 29 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|e^2=h,c^a=d,d^a=cdefg,e^a=egi,f^a=g,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 | #72267123, #72267156, #72268794, … |
| maximal normal subgroup of | #72267156, #72268794, C2×#14885⋊C3 |
| maximal quotient of | C2×#14885⋊C3, #72268821 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083667);