Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #88431 |
| automorphism | Aut(G) = C2×D4×((C3⋊D3)⋊C3)⋊C2 (order 1728) · Inn(G) = #6198 · Out(G) = C23 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| 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:
(2,8,6,4,5,9)(3,7)(10,11,13,15)(12,17,16,14);(1,2,4)(3,5,8,7,6,9)(11,14)(12,16)(15,17)Polycyclic presentation:
2^4,3^3|b^2=d,c^2=d,b^a=bc,c^a=cd,f^a=f^2,g^a=g^2,c^b=cd,e^b=e^2,g^b=g^2,f^e=fg29 conjugacy classes — view the full character table.
| maximal subgroup of | #11720837, #11720839, #11720844, … |
| maximal normal subgroup of | #11720837, #11720839, #11720844, … |
| maximal quotient of | #11720313, #11720314, #11720318, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 78);