Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C9×#456 ⋊ C2 |
| Sylow | C9 ⋊ #3291 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #10589692 |
| commutator, cosocle | C2×C4×C9 | C24 |
| Frattini | C2×C3×C4 | C23×D3 |
| Fitting | C9×#456 | C2 |
| socle | C2×C3 | #5590 |
| hypercenter | #456 | D9 |
| hypocenter | C9 | #3291 |
| LCS | G ▷ C2×C4×C9 ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #456 | |
690 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 690 conjugacy classes of subgroups — too many to draw. Only its 142 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:
2^7,3^2|a^2=g,e^2=g,f^2=g,h^3=i,b^a=be,c^a=cf,e^a=eg,f^a=fg,c^b=cg,d^b=de,e^b=eg,f^c=fg,e^d=eg,h^d=h^2i^2,i^d=i^293 conjugacy classes — view the full character table.
| maximal subgroup of | D9×#3291, #72268999, #72269000 |
| maximal normal subgroup of | D9×#3291, #72268999, #72269000 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 120848);