Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C9 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| derived, Sylow | #2840 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #5464 |
| commutator, hypocenter | #2840 | C9 |
| Frattini, socle | C23×C3 | C22⋊A4 |
| Fitting, cosocle | C3×#2840 | C3 |
| prime radical | C25×C3 | A4 |
| LCS | G ▷ #2840 | |
| UCS | C1 ◁ C2×C3 | |
247 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 247 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) = 72 |
Polycyclic presentation:
3^2,2^7|a^3=b,d^2=i,e^2=i,c^a=f,d^a=de,e^a=d,f^a=cf,g^a=h,h^a=gh,d^c=dg,e^c=eh,e^d=ei,f^d=fgh,f^e=fg42 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153666);