Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #13181783 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting, socle | C22×C32 ⋊ GL(2,𝔽3) |
| derived | C22⋊(PSU(3,𝔽2)⋊C3) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22⋊(PSU(3,𝔽2)⋊C3) | C2 |
| Fitting, socle | C22×C32 | GL(2,𝔽3) |
| LCS | G ▷ C22⋊(PSU(3,𝔽2)⋊C3) | |
| UCS | C1 | |
187 conjugacy classes of subgroups; only the 11 normal subgroups are shown.
This group has 187 conjugacy classes of subgroups — too many to draw. Only its 11 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:
2,3,2^5,3^2|c^2=g,d^2=g,b^a=b^2,c^a=dg,d^a=cg,e^a=ef,h^a=h^2i,c^b=d,d^b=cdg,e^b=f,f^b=ef,h^b=hi,d^c=dg,h^c=hi,i^c=hi^2,h^d=i^2,i^d=h,h^g=h^2,i^g=i^220 conjugacy classes — view the full character table.
| maximal subgroup of | #72213290, C2×#13181783 |
| maximal normal subgroup of | C2×#13181783 |
| maximal quotient of | #72220495, C2×#13181783 |
| automorphism group of | C2×PSU(3,𝔽2), C2×SU(3,𝔽2) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47846);