Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #3688⋊C3 |
| automorphism | Aut(G) = C2×((C22⋊C9)⋊C2)⋊C3 (order 432) · Inn(G) = ((C22⋊C9)⋊C2)⋊C3 · Out(G) = C2 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #3688⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | ((C22⋊C9)⋊C2)⋊C3 |
| commutator, cosocle, hypocenter | Q8⋊C9 | C2×C3 |
| Frattini, socle | C2×C3 | C3×S4 |
| Fitting | C32×Q8 | D3 |
| prime radical | C2×C32 | S4 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C2 | |
46 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,6,10,13,5,15,12,7)(2,11,14,4,8,3,16,9)(18,22,25,20,21,24)(19,23);(3,9,12)(4,10,11)(6,13,16)(7,14,15)(17,25,20,23,21,22,19,18,24)Polycyclic presentation:
2,3^3,2^3|a^2=g,c^3=d^2,e^2=g,f^2=g,c^a=c^2d,d^a=d^2,e^a=f,f^a=e,c^b=cd,e^c=fg,f^c=ef,f^e=fg29 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#3688⋊C3, #63689⋊C3, #63690⋊C3, … |
| maximal normal subgroup of | C2×#3688⋊C3, #63689⋊C3, #63690⋊C3, … |
| maximal quotient of | #63421⋊C3, C2×#3688⋊C3, #3688⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 243);