Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #13173046 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23×C33 |
No aliases for this group.
| Fitting | C32×#120 ⋊ D3 |
| derived | C32×C22⋊A4 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C32×C22⋊A4 | C22 |
| Frattini | C22 | C2×C3⋊(C3⋊S4) |
| Fitting | C32×#120 | D3 |
| socle | C22×C32 | C2×S4 |
| LCS | G ▷ C32×C22⋊A4 | |
| UCS | C1 | |
898 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 898 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^2,3^3,2^4|c^a=c^2,d^a=d^2,e^a=e^2,f^a=fg,h^a=hi,f^b=fh,g^b=gi,f^c=g,g^c=fg,h^c=i,i^c=hi54 conjugacy classes — view the full character table.
| maximal subgroup of | #72220693, C3⋊D3×#1020⋊C2, D3×#10593630 |
| maximal normal subgroup of | C3⋊D3×#1020⋊C2, D3×#10593630 |
| maximal quotient of | #72220693 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 39109);