Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #84125⋊C3 |
| Fitting | C3×#585 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C3⋊(#120⋊C3) |
| commutator, cosocle, hypocenter | C3×H(5,𝔽2) | C22×C3 |
| Frattini | C23 | C2×D3×A4 |
| Fitting | C3×#585 | C2×C3 |
| socle | C2×C3 | C2×#120⋊C3 |
| LCS | G ▷ C3×H(5,𝔽2) | |
| UCS | C1 ◁ C4 | |
360 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 360 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) = 48 |
Polycyclic presentation:
2^2,3,2^4,3,2|b^2=i,d^a=df,e^a=efg,h^a=h^2,d^c=de,e^c=d,f^c=gi,g^c=fgi,g^d=gi,f^e=fi,g^e=gi54 conjugacy classes — view the full character table.
| maximal subgroup of | #72274864, #72274884, #72274889, … |
| maximal normal subgroup of | #72274864, #72274884, #72274889, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 155394);