Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #80308⋊C3 |
| Fitting | C3×#458 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D3×#120⋊C3 |
| commutator, cosocle, hypocenter | C3×H(5,𝔽2) | C22×C3 |
| Frattini | C23 | C2×D3×A4 |
| Fitting | C3×#458 | C2×C3 |
| socle | C2×C3 | C2×#120⋊C3 |
| LCS | G ▷ C3×H(5,𝔽2) | |
| UCS | C1 ◁ C2 | |
353 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 353 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^2,3,2^4,3,2|a^2=i,b^2=i,h^a=h^2,d^b=df,e^b=efg,d^c=de,e^c=d,f^c=gi,g^c=fgi,g^d=gi,f^e=fi,g^e=gi48 conjugacy classes — view the full character table.
| maximal subgroup of | #72274907, #72274911, #72274920, … |
| maximal normal subgroup of | #72274907, #72274911, #72274920, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 155240);