Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of ((C3⋊S4)⋊A4)⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | ((C3⋊S4)⋊A4)⋊C2 |
| derived | (C3×A4)⋊A4 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | (C3×A4)⋊A4 | C22 |
| Frattini | C3 | S42 |
| Fitting, socle | C24×C3 | D32 |
| LCS | G ▷ (C3×A4)⋊A4 | |
| UCS | C1 | |
545 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 545 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 54 |
Polycyclic presentation:
2^2,3^3,2^4|c^a=c^2,e^a=e^2,h^a=hi,d^b=d^2,e^b=e^2,f^b=fg,d^c=de,h^c=i,i^c=hi,f^d=g,g^d=fg35 conjugacy classes — view the full character table.
| maximal subgroup of | C3×((C3⋊S4)⋊A4)⋊C2, #72222956 |
| maximal normal subgroup of | C3×((C3⋊S4)⋊A4)⋊C2 |
| maximal quotient of | C3×((C3⋊S4)⋊A4)⋊C2 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46099);