Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #1437⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2×C4 |
| unambiguous semidirect product | #7554⋊C3 |
| Fitting, Sylow | #7554 ⋊ C3 |
| derived | C22×C42 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #1437⋊C3 |
| commutator, hypocenter | C22×C42 | C3×C4 |
| Frattini, socle | C25 | C2×A4 |
| Fitting | #7554 | C3 |
| prime radical | C2×C22⋊(C42⋊C3) | C2 |
| cosocle | C23×C42 | C2×C3 |
| LCS | G ▷ C22×C42 | |
| UCS | C1 ◁ C2 | |
573 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 573 conjugacy classes of subgroups — too many to draw. Only its 22 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|b^2=g,c^2=h,d^2=i,b^a=behi,c^a=cdef,d^a=cfi,e^a=ef,f^a=e,h^a=hi,i^a=h,c^b=ce,d^b=df48 conjugacy classes — view the full character table.
| maximal subgroup of | #72237333, #72237338, C4×#1437⋊C3, … |
| maximal normal subgroup of | #72237338, C4×#1437⋊C3, #72267794, … |
| maximal quotient of | #72267335, #72267795 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083480);