Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of C3⋊#85559 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22×C3×C4 |
| unambiguous semidirect product | C3⋊#85559 |
| Fitting | C3×H(3,𝔽4) ⋊ D3 |
| derived | C3×#5464 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×#5464 | C2 |
| Frattini | C22 | C3⋊#1020 |
| Fitting | C3×H(3,𝔽4) | D3 |
| socle | C22×C3 | #1020 |
| LCS | G ▷ C3×#5464 | |
| UCS | C1 | |
286 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 286 conjugacy classes of subgroups — too many to draw. Only its 17 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:
2,3^2,2^6|b^a=b^2,c^a=c^2,d^a=de,f^a=fg,h^a=hi,d^b=e,e^b=de,f^b=g,g^b=fg,h^b=i,i^b=hi,f^d=fh,g^d=ghi,f^e=fhi,g^e=gi21 conjugacy classes — view the full character table.
| maximal subgroup of | #72215162, D3×#85559, #72215235, … |
| maximal normal subgroup of | #72215162, D3×#85559, #72215235 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157139);