Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C9 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #1780⋊C9 |
| Sylow | #1780 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | #120⋊C3 |
| commutator, hypocenter | C22×Q8 | C4×C9 |
| Frattini | C22×C3×C4 | C2×A4 |
| Fitting | C3×#1780 | C3 |
| socle | C23×C3 | C4×A4 |
| prime radical, cosocle | C22×C3×#41 | C2×C3 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C3×C4 | |
239 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 239 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2,3,2,3,2^5|a^2=c,b^3=d,c^2=i,e^2=i,f^2=i,e^a=eg,f^a=fh,e^b=f,f^b=efi,g^b=h,h^b=gh,f^e=fi108 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153358);