Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #88422 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C3×C9 ⋊ Q16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #6195 |
| commutator, Fitting, cosocle | C3×C4×C9 | C22 |
| Frattini | C3×C4 | D32 |
| socle, prime radical | C2×C32 | #67 |
| hypercenter | C4 | D3×D9 |
| hypocenter | C3×C9 | Q16 |
| LCS | G ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C2 ◁ C4 | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,3)(5,7)(6,10)(8,9)(11,12)(13,15,22,26,17,24,16,23)(14,18,25,28,20,27,19,21);(2,3)(4,5,8,10,12,11,6,9,7)(13,14,17,20)(15,21,24,28)(16,25,22,19)(18,26,27,23)Polycyclic presentation:
2^4,3^3|a^2=d,b^2=cd,c^2=d,e^3=g^2,b^a=bc,c^a=cd,f^a=f^2,e^b=e^2g,g^b=g^251 conjugacy classes — view the full character table.
| maximal subgroup of | #11720821, D9×Dic12, #11720827, … |
| maximal normal subgroup of | #11720821, D9×Dic12, #11720827, … |
| maximal quotient of | #11720262, #11720269, #11720298, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 75);