Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | C3⋊#68001 |
| derived | C3×C22⋊SL(2,𝔽3) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊#4625 |
| commutator, hypocenter | C3×C22⋊SL(2,𝔽3) | C4 |
| Frattini | C24 | C3⋊S4 |
| Fitting | C3×#393 | D3 |
| socle | C23×C3 | SL(2,ℤ/4) |
| prime radical, cosocle | C3×#393⋊C3 | C2 |
| LCS | G ▷ C3×C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
184 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 27 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:
2^2,3^2,2^5|a^2=b,e^2=i,f^2=i,c^a=c^2,d^a=d^2,e^a=efg,f^a=fghi,g^a=gh,e^b=eg,f^b=fh,e^c=f,f^c=efi,g^c=h,h^c=gh,f^e=fi39 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 154148);