Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C9 · Schur multiplier (H2) = C23×C4 |
| unambiguous semidirect product | #2679⋊C9 |
| derived, Sylow | #2679 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #5464 |
| commutator, hypocenter | #2679 | C9 |
| Frattini, socle | C23×C3 | C22⋊A4 |
| Fitting, prime radical, cosocle | C3×#2679 | C3 |
| LCS | G ▷ #2679 | |
| UCS | C1 ◁ C2×C3 | |
401 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 401 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
3^2,2^7|a^3=b,c^2=g,d^2=gh,c^a=cd,d^a=cgh,e^a=ef,f^a=e,g^a=h,h^a=gh,i^a=ghi,e^c=ehi,f^c=fg,e^d=ei,f^d=fghi48 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153656);