Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of C3⋊(C3⋊#226) |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C33 |
| unambiguous semidirect product | C3⋊(C3⋊#226) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊(C3⋊S4) |
| commutator, prime radical, cosocle, hypocenter | C32×SL(2,𝔽3) | C2 |
| Fitting | C32×Q8 | D3 |
| socle | C2×C32 | S4 |
| LCS | G ▷ C32×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,6,10,13,5,15,12,7)(2,11,14,4,8,3,16,9)(18,19)(21,22);(20,21,22);(17,18,19);(3,9,12)(4,10,11)(6,13,16)(7,14,15)Polycyclic presentation:
2,3^3,2^3|a^2=g,e^2=g,f^2=g,b^a=b^2,c^a=c^2,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 | #11724176, #11724177, C3⋊D3×#226, … |
| maximal normal subgroup of | #11724176, #11724177, C3⋊D3×#226, … |
| maximal quotient of | C3⋊(C3⋊#859), C2×C3⋊(C3⋊#226), C3⋊(C9⋊#226), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 619);