Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #15258⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22×C4 |
| unambiguous semidirect product | #15258⋊C3 |
| Fitting, Sylow | #15258 ⋊ C3 |
| derived | C22×C42 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22×C42 | C22×C3 |
| Frattini, socle | C24 | C22×A4 |
| Fitting | #15258 | C3 |
| LCS | G ▷ C22×C42 | |
| UCS | C1 | |
929 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 929 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) = 24 |
Polycyclic presentation:
3,2^8|c^2=f,b^a=bcfghi,c^a=befgh,d^a=dfgi,e^a=ehi,f^a=i,g^a=gh,h^a=g,i^a=fi,c^b=cf,d^b=dg,e^b=ei,d^c=dh,e^c=ef30 conjugacy classes — view the full character table.
| maximal subgroup of | #72209644, #72214108, #72219762, … |
| maximal normal subgroup of | #72209644, #72219762, C2×#15258⋊C3, … |
| maximal quotient of | #72209777, #72209788, #72226624, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083691);