Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C3×C8 |
No aliases for this group.
| Fitting | C32×C82 ⋊ C2 |
| Sylow | C32 ⋊ #1712 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #64083 |
| commutator, cosocle | C32×C42 | C23 |
| Frattini | C42 | C22×C3⋊D3 |
| Fitting | C32×C82 | C2 |
| socle | C22×C32 | #127 |
| hypercenter | C82 | C3⋊D3 |
| hypocenter | C32 | #1712 |
| LCS | G ▷ C32×C42 ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 | |
840 conjugacy classes of subgroups.
| transitive degree | μt(G) = 576 |
Polycyclic presentation:
2^7,3^2|b^2=df,c^2=eg,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^a=h^2,i^a=i^2294 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊D3×#1712, #72214093, #72214125, … |
| maximal normal subgroup of | C3⋊D3×#1712, #72214093, #72214125, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 16599);