Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #8530⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2×C4 |
| unambiguous semidirect product | #8530⋊C3 |
| Fitting, Sylow | #8530 ⋊ C3 |
| derived | H(3,𝔽4) ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | H(3,𝔽4) | C3×C4 |
| Frattini | C2×C42 | C2×A4 |
| Fitting | #8530 | C3 |
| socle | C22 | #379⋊C3 |
| prime radical | #3549⋊C3 | C2 |
| cosocle | #3549 | C2×C3 |
| LCS | G ▷ H(3,𝔽4) | |
| UCS | C1 | |
178 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 12 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
3,2^8|b^2=g,e^2=i,f^2=h,g^2=hi,b^a=bch,c^a=cdfgi,d^a=cgh,e^a=ef,f^a=eh,g^a=egi,h^a=i,i^a=hi,c^b=ce,d^b=df,e^b=eh,f^b=fi,e^c=ei,f^c=fhi,g^c=ghi,e^d=ehi,f^d=fh,g^d=ghi24 conjugacy classes — view the full character table.
| maximal subgroup of | #72209647, #72272694, #72340791, … |
| maximal normal subgroup of | #72209647, #72272694, #72340797, … |
| maximal quotient of | #72209726, #72209797, #72209814 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083510);