Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #6249⋊C2 |
| automorphism | Aut(G) = #6249⋊C2 (order 432) · Inn(G) = G · Out(G) = C1 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #6249⋊C2 |
| unambiguous semidirect product | SU(3,𝔽2)⋊C2 |
| Fitting, Sylow | H(3,𝔽3) ⋊ SD16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle | #1088 | C22 |
| Frattini, socle | C3 | AΓL(1,𝔽9) |
| Fitting, hypocenter | H(3,𝔽3) | SD16 |
| prime radical | #1088⋊C2 | C2 |
| LCS | G ▷ #1088 ▷ #272 ▷ H(3,𝔽3) | |
| UCS | C1 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,9,21,4,27,17)(2,8,26,19,18,24)(3,25,23,13,6,20)(5,22,15,10,7,11)(12,16,14);(2,3,6,12,7,13,20,14)(4,16)(5,17,8,11,18,9,10,24)(15,19,21,23,26,22,27,25)Polycyclic presentation:
2^4,3^3|a^2=c,c^2=d,b^a=bcd,e^a=f,f^a=ef^2,g^a=g^2,c^b=cd,e^b=ef^2g,f^b=f^2,g^b=g^2,e^c=ef^2,f^c=e^2f^2g,e^d=e^2g^2,f^d=f^2g,f^e=fg14 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 520);