Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C3⋊(#2202⋊C3) |
| unambiguous semidirect product | #71951⋊C3 |
| Fitting, derived | C3×H(3,𝔽4) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C3⋊(B(4,𝔽2)⋊C3) |
| commutator, Fitting, cosocle, hypocenter | C3×H(3,𝔽4) | C2×C3 |
| Frattini | C2×Q8 | D3×A4 |
| socle | C2×C3 | B(4,𝔽2)⋊C3 |
| hypercenter | C22 | C3⋊(#120⋊C3) |
| LCS | G ▷ C3×H(3,𝔽4) | |
| UCS | C1 ◁ C2 ◁ C22 | |
178 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 15 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:
2,3,2^4,3,2^2|e^2=i,f^2=i,c^a=ce,d^a=defi,e^a=ei,g^a=g^2,h^a=hi,c^b=cd,d^b=c,e^b=fh,f^b=efhi,e^c=ei,f^c=fh,e^d=eh,f^d=fhi,f^e=fi36 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 154185);