Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| derived | C32⋊(C22⋊A4) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C3 | C3⋊#1020 |
| commutator, hypocenter | C32⋊(C22⋊A4) | C4 |
| Fitting | C25×C32 | D3 |
| socle | C25×C3 | C3⋊D3 |
| prime radical, cosocle | C2×C32⋊(C22⋊A4) | C2 |
| LCS | G ▷ C32⋊(C22⋊A4) | |
| UCS | C1 ◁ C2×C3 | |
771 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 771 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2^2,3^3,2^4|a^2=b,c^a=c^2,d^a=d^2,f^a=fg,h^a=hi,d^c=de,f^c=g,g^c=fg,h^c=i,i^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72242699 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30623);