Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #6249 |
| automorphism | Aut(G) = #6249⋊C4 (order 864) · Inn(G) = #6249 · Out(G) = C4 |
| homology | Abelianization (H1) = C16 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | H(3,𝔽3) ⋊ C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #6249 |
| commutator, hypocenter | H(3,𝔽3) | C16 |
| Frattini, socle | C2×C3 | AGL(1,𝔽9) |
| Fitting, prime radical | C2×H(3,𝔽3) | C8 |
| cosocle | #6188 | C2 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C2 | |
21 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 43 points:
(2,3,5,10,6,11,16,12)(4,15)(7,13,19,8,21,14,26,23)(9,25,18,17,20,27,22,24)(28,29,30,33,31,34,36,39,32,35,37,40,38,41,42,43);(1,18,22)(2,17,3)(4,26,19)(5,16,15)(6,11,24)(7,27,8)(9,10,14)(12,20,13)(21,23,25)Polycyclic presentation:
2^4,3^3|a^2=b,b^2=c,c^2=d,e^a=f,f^a=ef^2,g^a=g^2,e^b=ef^2,f^b=e^2f^2g^2,e^c=e^2g,f^c=f^2g^2,f^e=fg^226 conjugacy classes — view the full character table.
| maximal subgroup of | #11722287, #11722288, C2×#88586, … |
| maximal normal subgroup of | #11722287, #11722288, C2×#88586, … |
| maximal quotient of | #11720689, C2×#88586, C3×#88586, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 233);