Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3×C9 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D3×D9 |
| commutator, cosocle | C2×C3×C9 | C23 |
| Frattini | C2×C3 | C2×D32 |
| Fitting | C3×C4×C9 | C22 |
| socle | C2×C32 | C22×D3 |
| hypercenter | C4 | D3×D9 |
| hypocenter | C3×C9 | #41 |
| LCS | G ▷ C2×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C2 ◁ C4 | |
132 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3)(4,9,11,10,5,7,6,12,8)(13,15,16,19)(14,18,17,20);(5,7)(6,10)(8,9)(11,12)(14,17)(15,19);(2,3)(13,14)(15,18)(16,17)(19,20)Polycyclic presentation:
2^4,3^3|c^2=d,e^3=g^2,b^a=bd,f^a=f^2,c^b=cd,e^b=e^2g,g^b=g^263 conjugacy classes — view the full character table.
| maximal subgroup of | #11720796, #11720805, #11720810, … |
| maximal normal subgroup of | #11720796, #11720805, #11720810, … |
| maximal quotient of | #11720906, #11720912, #11720919, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 288);