Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C9 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #1790⋊C9 |
| derived | H(5,𝔽2) ⋊ C4×C9 |
| Sylow | #1790 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22×C3 | #120⋊C3 |
| commutator, hypocenter | H(5,𝔽2) | C4×C9 |
| Frattini | C24×C3 | C2×A4 |
| Fitting | C3×#1790 | C3 |
| prime radical, cosocle | C2×C3×H(5,𝔽2) | C2×C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C22×C3 | |
284 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2,3,2,3,2^5|a^2=c,b^3=d,e^a=eg,f^a=fh,e^b=f,f^b=ef,g^b=h,h^b=ghi,h^e=hi,g^f=gi96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153360);