Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #1088⋊C2 |
| automorphism | Aut(G) = C22×#1088⋊C2 (order 864) · Inn(G) = #1088⋊C2 · Out(G) = C22 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | (((C3⋊D3)⋊C3)⋊C2)⋊C4 |
| Sylow | H(3,𝔽3) ⋊ C22⋊C4 |
99 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,3,13,11,5,12)(2,6,15)(4,10)(7,9,8,16,18,14)(19,20,21,22);(1,2)(3,9)(4,7)(5,14)(6,10)(8,13)(11,18)(12,15)(16,17)Polycyclic presentation:
2^4,3^3|b^2=d,b^a=bc,e^a=e^2,g^a=g^2,e^b=f,f^b=e,g^b=g^2,e^c=e^2,f^c=f^2,f^e=fg26 conjugacy classes — view the full character table.
| maximal subgroup of | (((C3⋊D3)⋊C3)⋊C2)⋊Q8, #11722302, C4×#1088⋊C2, … |
| maximal normal subgroup of | (((C3⋊D3)⋊C3)⋊C2)⋊Q8, #11722302, C4×#1088⋊C2, … |
| maximal quotient of | #11720694, #11720695, #11720696, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 238);