Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22×C4 |
| unambiguous semidirect product | #37611⋊C3 |
| Fitting, Sylow | #37611 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #3549⋊C3 |
| commutator, cosocle, hypocenter | #2844 | C2×C3 |
| Frattini, socle | C23 | C2×C22⋊A4 |
| Fitting | #37611 | C3 |
| LCS | G ▷ #2844 | |
| UCS | C1 ◁ C2 | |
343 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 343 conjugacy classes of subgroups — too many to draw. Only its 20 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=g,c^2=g,e^2=ghi,f^2=h,b^a=bcde,c^a=cefgh,d^a=bcefi,f^a=ce,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084440);