Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #8101⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #8101⋊C3 |
| Fitting, Sylow | #8101 ⋊ C3 |
| derived | #1304 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #1304 | C2×C3 |
| Frattini | C25 | C2×A4 |
| Fitting | #8101 | C3 |
| socle | C22 | C2×#95⋊C3 |
| LCS | G ▷ #1304 | |
| UCS | C1 | |
464 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 464 conjugacy classes of subgroups — too many to draw. Only its 15 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
Polycyclic presentation:
3,2^8|b^2=f,c^2=g,b^a=cefi,c^a=bcdefghi,d^a=degi,e^a=ehi,f^a=ghi,g^a=efghi,h^a=i,i^a=hi,c^b=ce,d^b=dh,e^b=eh,g^b=gi,e^c=ei,f^c=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | #72210409, #72210571, #72226619, … |
| maximal normal subgroup of | #72210571, C2×#8101⋊C3 |
| maximal quotient of | #72227397, C2×#8101⋊C3, #72234947, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083496);