Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22×C4 |
| unambiguous semidirect product | #35438⋊C3 |
| Fitting, Sylow | #35438 ⋊ C3 |
| derived | #2273 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #3444⋊C3 |
| commutator, cosocle, hypocenter | #2273 | C2×C3 |
| Frattini, socle | C23 | C2×C22⋊A4 |
| Fitting | #35438 | C3 |
| LCS | G ▷ #2273 | |
| UCS | C1 ◁ C2 | |
457 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 457 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
3,2^8|c^2=g,e^2=g,f^2=h,c^a=df,d^a=cdehi,e^a=fg,f^a=ef,g^a=h,h^a=gh,c^b=cg,d^b=dh,d^c=di,f^c=fi,e^d=ei32 conjugacy classes — view the full character table.
| maximal subgroup of | #72227365, #72239009, #72341122 |
| maximal normal subgroup of | #72227365, #72341122 |
| maximal quotient of | #72222710 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084264);