Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of C9⋊#226 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C9⋊#226 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C9⋊S4 |
| commutator, cosocle, hypocenter | C9×SL(2,𝔽3) | C2 |
| Frattini, socle | C2×C3 | C3⋊S4 |
| Fitting | Q8×C9 | D3 |
| prime radical | C3×SL(2,𝔽3) | D3 |
| LCS | G ▷ C9×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
59 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,6,10,13,5,15,12,7)(2,11,14,4,8,3,16,9)(18,20)(19,23)(21,22)(24,25);(17,25,20,23,22,21,19,18,24);(3,9,12)(4,10,11)(6,13,16)(7,14,15)Polycyclic presentation:
2,3^3,2^3|a^2=g,c^3=d^2,e^2=g,f^2=g,b^a=b^2,c^a=c^2d,d^a=d^2,e^a=f,f^a=e,e^b=fg,f^b=ef,f^e=fg36 conjugacy classes — view the full character table.
| maximal subgroup of | D9×#226, #11722231, #11722233, … |
| maximal normal subgroup of | D9×#226, #11722231, #11722233, … |
| maximal quotient of | C9⋊#859, C2×C9⋊#226, C27⋊#226, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 252);