Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #88421 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3×C9 ⋊ D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #6194 |
| commutator, Fitting, cosocle | C3×C4×C9 | C22 |
| Frattini | C3×C4 | D32 |
| socle | C2×C32 | #67 |
| hypercenter | C4 | D3×D9 |
| hypocenter | C3×C9 | D8 |
| LCS | G ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C2 ◁ C4 | |
74 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,3)(5,7)(6,10)(8,9)(11,12)(14,17)(15,20)(18,19);(2,3)(4,5,8,10,12,11,6,9,7)(13,14)(15,17)(16,19)(18,20)Polycyclic presentation:
2^4,3^3|c^2=d,e^3=g^2,b^a=bc,c^a=cd,f^a=f^2,c^b=cd,e^b=e^2g,g^b=g^248 conjugacy classes — view the full character table.
| maximal subgroup of | #11720818, #11720819, #11720820, … |
| maximal normal subgroup of | #11720818, #11720819, #11720820, … |
| maximal quotient of | #11720238, #11720240, #11720242, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 68);