Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #3754⋊C3 |
| automorphism | Aut(G) = C2×D4×((C3⋊D3)⋊C3)⋊C2 (order 1728) · Inn(G) = #638⋊C3 · Out(G) = C23 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | #3756⋊C3 |
| Sylow | H(3,𝔽3) ⋊ Q16 |
75 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,18,24)(19,21,23)(20,22,25);(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(18,20)(19,22)(21,23)(24,25)Polycyclic presentation:
2^2,3,2^2,3^2|a^2=e,b^2=e,d^2=e,b^a=bd,d^a=de,f^a=f^2,g^a=g^2,d^b=de,f^c=fg41 conjugacy classes — view the full character table.
| maximal subgroup of | #11721226, #11721227, #11721230, … |
| maximal normal subgroup of | #11721226, #11721227, #11721230, … |
| maximal quotient of | #63634⋊C3, #63637⋊C3, #63662⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 160);