Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #3746⋊C3 |
| automorphism | Aut(G) = C2×D4×((C3⋊D3)⋊C3)⋊C2 (order 1728) · Inn(G) = C4⋊((C3⋊D3)⋊C3) · Out(G) = C23 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | #3744⋊C3 |
| Fitting | C8×H(3,𝔽3) ⋊ C2 |
| Sylow | H(3,𝔽3) ⋊ SD16 |
78 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,3,5,4,6,7,8)(9,10,16)(11,13,15)(12,14,17);(2,5)(3,7)(6,8)(10,12)(11,14)(13,15)(16,17)Polycyclic presentation:
2^2,3,2^2,3^2|b^2=d,d^2=e,b^a=bd,d^a=de,f^a=f^2,g^a=g^2,f^c=fg53 conjugacy classes — view the full character table.
| maximal subgroup of | #11720837, #11720839, #11720843, … |
| maximal normal subgroup of | #11720837, #11720839, #11720843, … |
| maximal quotient of | #63643⋊C3, #63644⋊C3, #63648⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 117);