Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C9×#444 ⋊ C2 |
| Sylow | C9 ⋊ #3288 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×D36 |
| commutator, cosocle | C2×C4×C9 | C24 |
| Frattini | C2×C3×C4 | C23×D3 |
| Fitting | C9×#444 | C2 |
| socle | C2×C3 | D4×D12 |
| hypercenter | #444 | D9 |
| hypocenter | C9 | #3288 |
| LCS | G ▷ C2×C4×C9 ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #444 | |
696 conjugacy classes of subgroups.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^7,3^2|e^2=g,f^2=g,h^3=i,b^a=be,c^a=cf,e^a=eg,f^a=fg,h^a=h^2i^2,i^a=i^2,d^b=de,e^b=eg,f^c=fg,h^c=h^2i^2,i^c=i^2,e^d=eg114 conjugacy classes — view the full character table.
| maximal subgroup of | #72268966, D9×#3288, #72269012 |
| maximal normal subgroup of | #72268966, D9×#3288, #72269012 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 120585);