Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #1088 |
| automorphism | Aut(G) = C4×AΓL(1,𝔽9) (order 576) · Inn(G) = #157 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C16 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | H(3,𝔽3) ⋊ C16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3×C4 | #157 |
| commutator, hypocenter | H(3,𝔽3) | C16 |
| Fitting | C4×H(3,𝔽3) | C4 |
| socle | C2×C3 | #622 |
| prime radical | C2×H(3,𝔽3) | C8 |
| cosocle | #6180 | C2 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C3×C4 | |
31 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 34 points:
(1,2)(3,7,9,14)(4,11)(5,8)(6,12,13,18)(10,15,16,17)(19,20,21,24,22,25,27,30,23,26,28,31,29,32,33,34);(1,3,16)(2,15,6)(4,12,9)(5,10,18)(7,17,11)(8,13,14)Polycyclic presentation:
2^4,3^3|a^2=b,b^2=c,c^2=d,e^a=ef^2,f^a=e^2f^2,e^b=e^2,f^b=f^2g,f^e=fg56 conjugacy classes — view the full character table.
| maximal subgroup of | #11720689, #11720690, #11720691, … |
| maximal normal subgroup of | #11720689, #11720690, #11720691, … |
| maximal quotient of | #11720227, C2×#88410, C3×#88410, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 57);