Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #13378⋊C3 |
| Fitting, Sylow | #13378 ⋊ C3 |
| derived | #2811 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2)⋊C3 |
| commutator, cosocle, hypocenter | #2811 | C2×C3 |
| Frattini | C22×Q8 | C2×A4 |
| Fitting | #13378 | C3 |
| hypercenter | C23 | #120⋊C3 |
| LCS | G ▷ #2811 | |
| UCS | C1 ◁ C22 ◁ C23 | |
187 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 187 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 32 |
Polycyclic presentation:
3,2^8|b^2=e,c^2=e,f^2=i,g^2=i,b^a=bcei,c^a=bhi,d^a=dg,f^a=fg,g^a=fhi,c^b=ce,d^b=df,f^b=fi,g^b=ghi,d^c=dg,f^c=fh,g^c=gi,f^d=fi,g^d=gi,h^d=hi,g^f=gi37 conjugacy classes — view the full character table.
| maximal subgroup of | #72209606, #72227401, #72227417, … |
| maximal normal subgroup of | #72227858, #72227859, #72227888, … |
| maximal quotient of | #72238090, #72267022, #72267519, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083573);