Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #6196 |
| automorphism | Aut(G) = C24×((C3⋊D3)⋊C3)⋊C2 (order 1728) · Inn(G) = ((C3⋊D3)⋊C3)⋊C2 · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | H(3,𝔽3) ⋊ C22⋊C4 |
135 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 135 conjugacy classes of subgroups — too many to draw. Only its 35 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 — 2 generators on 17 points:
(2,8,6,4,5,9)(3,7)(10,11,12,13)(15,17);(1,2,4)(3,5,8,7,6,9)(14,15)(16,17)Polycyclic presentation:
2^4,3^3|b^2=c,b^a=bd,f^a=f^2,g^a=g^2,e^b=e^2,g^b=g^2,f^e=fg38 conjugacy classes — view the full character table.
| maximal subgroup of | #11721081, #11721100, #11721101, … |
| maximal normal subgroup of | #11721081, #11721100, #11721101, … |
| maximal quotient of | #11720305, #11720308, #11720309, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 97);