Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | C3⋊#68003 |
| derived | C3×#95⋊C3 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊#4626 |
| commutator, hypocenter | C3×#95⋊C3 | C4 |
| Frattini | C24 | C3⋊S4 |
| Fitting | C3×#399 | D3 |
| socle | C23×C3 | SL(2,ℤ/4) |
| prime radical, cosocle | C3×#399⋊C3 | C2 |
| LCS | G ▷ C3×#95⋊C3 | |
| UCS | C1 ◁ C2 | |
176 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 176 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) = 72 |
Polycyclic presentation:
2^2,3^2,2^5|a^2=b,e^2=g,f^2=ghi,c^a=c^2,d^a=d^2,e^a=efhi,f^a=fg,g^a=gi,h^a=hi,e^b=eh,f^b=fi,e^c=fh,f^c=efgh,g^c=ghi,h^c=i,i^c=hi,f^e=fgh39 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 154150);