Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22×C4 |
| unambiguous semidirect product | #37563⋊C3 |
| Fitting, Sylow | #37563 ⋊ C3 |
| derived | #2844 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #3568⋊C3 |
| commutator, cosocle, hypocenter | #2844 | C2×C3 |
| Frattini, socle | C23 | C2×C22⋊A4 |
| Fitting | #37563 | C3 |
| LCS | G ▷ #2844 | |
| UCS | C1 ◁ C2 | |
280 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 24 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=i,d^2=h,f^2=g,b^a=bdfi,c^a=cefg,d^a=bcegi,e^a=egi,f^a=ceghi,g^a=i,h^a=hi,i^a=gi,c^b=cg,d^b=dh,e^b=eg,f^b=fgi,d^c=di,e^d=egi28 conjugacy classes — view the full character table.
| maximal subgroup of | #72238835 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084436);