Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #89092 |
| automorphism | Aut(G) = D3×AΓL(1,𝔽9) (order 864) · Inn(G) = G · Out(G) = C2 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, Sylow | C33 ⋊ SD16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle | C3×#157 | C22 |
| Fitting, socle, hypocenter | C33 | SD16 |
| prime radical | #157⋊D3 | C2 |
| LCS | G ▷ C3×#157 ▷ C3×C3⋊D3 ▷ C33 | |
| UCS | C1 | |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,2,3)(4,5,8,7)(6,9,10,11);(2,3)(6,11)(7,10)(8,12)Polycyclic presentation:
2^4,3^3|b^2=d,c^2=d,b^a=bc,c^a=cd,f^a=e^2f^2,g^a=g^2,c^b=cd,e^b=f^2,f^b=e,e^c=ef^2,f^c=e^2f^2,e^d=e^2,f^d=f^215 conjugacy classes — view the full character table.
| maximal subgroup of | D3×AΓL(1,𝔽9), C2×#89092, (PSU(3,𝔽2)⋊C9)⋊C2, … |
| maximal normal subgroup of | D3×AΓL(1,𝔽9), C2×#89092, C3×#89092 |
| maximal quotient of | #11724101, #11724102, #11724103, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 739);