Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| derived | SL(2,𝔽3)⋊(C22⋊C9) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A4⋊((C22⋊C9)⋊C2) |
| commutator, cosocle, hypocenter | SL(2,𝔽3)⋊(C22⋊C9) | C2 |
| Frattini, socle | C2×C3 | A4⋊S4 |
| Fitting | C3×H(5,𝔽2) | C3⋊D3 |
| LCS | G ▷ SL(2,𝔽3)⋊(C22⋊C9) | |
| UCS | C1 ◁ C2 | |
227 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 227 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2,3^3,2^5|c^3=d,e^2=i,f^2=i,g^2=i,h^2=i,b^a=b^2,c^a=c^2d^2,d^a=d^2,e^a=ef,f^a=fi,g^a=gh,h^a=hi,g^b=h,h^b=ghi,e^c=f,f^c=efi,f^e=fi,h^g=hi45 conjugacy classes — view the full character table.
| maximal subgroup of | #72252971, C2×#13164076 |
| maximal normal subgroup of | #72252971, C2×#13164076 |
| maximal quotient of | #72245426, C2×#13164076 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30139);