Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2×C3 | C3⋊#1020 |
| commutator, prime radical, cosocle, hypocenter | C32⋊#997 | C2 |
| Fitting | C32×H(5,𝔽2) | D3 |
| LCS | G ▷ C32⋊#997 | |
| UCS | C1 ◁ C2×C3 | |
330 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2,3^3,2^5|a^2=i,g^2=i,h^2=i,b^a=b^2,c^a=c^2,e^a=ef,g^a=gh,h^a=hi,c^b=cd,e^b=f,f^b=ef,g^b=h,h^b=ghi,h^e=hi,g^f=gi,h^g=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72242719 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30619);