Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #634⋊C3 |
| automorphism | Aut(G) = C2×D4×((C3⋊D3)⋊C3)⋊C2 (order 1728) · Inn(G) = C2×(C3⋊D3)⋊C3 · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | #3748⋊C3 |
| Sylow | H(3,𝔽3) ⋊ M16 |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 17 points:
(2,5)(6,8)(9,10,16)(11,13,15)(12,14,17);(1,2,3,6,4,5,7,8)(10,12)(11,14)(13,15)(16,17)Polycyclic presentation:
2^2,3,2^2,3^2|a^2=d,d^2=e,b^a=be,f^a=f^2,g^a=g^2,f^c=fg62 conjugacy classes — view the full character table.
| maximal subgroup of | #11720305, #11720316, #11720319, … |
| maximal normal subgroup of | #11720305, #11720316, #11720319, … |
| maximal quotient of | #63630⋊C3, C4⋊((C3⋊(C3⋊C8))⋊C3), #63657⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 137);