Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of C3⋊(#1655⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2×C4 |
| unambiguous semidirect product | C3⋊(#1655⋊C3) |
| unambiguous semidirect product | #71948⋊C3 |
| Fitting, derived | C3×H(3,𝔽4) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C3×H(3,𝔽4) | C2×C3 |
| Frattini | C42 | D3×A4 |
| socle | C22×C3 | #120⋊C3 |
| LCS | G ▷ C3×H(3,𝔽4) | |
| UCS | C1 | |
150 conjugacy classes of subgroups; only the 13 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 13 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:
2,3,2^4,3,2^2|e^2=i,f^2=h,c^a=ce,d^a=defh,e^a=ei,f^a=fh,g^a=g^2,c^b=cd,d^b=c,e^b=fi,f^b=ef,h^b=hi,i^b=h,e^c=ei,f^c=fhi,e^d=eh,f^d=fi21 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 154184);