Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| automorphism | Aut(G) = C2×D4×((C3⋊D3)⋊C3)⋊C2 (order 1728) · Inn(G) = C2×((C3⋊D3)⋊C3)⋊C2 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | ((C3⋊D3)⋊C3)⋊Q8 |
| Sylow | H(3,𝔽3) ⋊ C2×Q8 |
145 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 145 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,9)(3,6,4)(5,8,7)(10,12,13,17)(11,14,15,16);(2,4)(3,7)(5,8)(6,9)(10,11,13,15)(12,16,17,14);(3,7)(5,6)(8,9)Polycyclic presentation:
2^4,3^3|b^2=d,c^2=d,f^a=f^2,g^a=g^2,c^b=cd,e^b=e^2,g^b=g^2,f^e=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #11720846, #11720851, #11720852, … |
| maximal normal subgroup of | #11720846, #11720851, #11720852, … |
| maximal quotient of | #11721077, #11721079, #11721080, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 294);