Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Fitting | Q8×H(3,𝔽3) ⋊ C2 |
| Sylow | H(3,𝔽3) ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×(C3⋊D3)⋊C3 |
| commutator, cosocle | C2×C32 | C23×C3 |
| Frattini, socle | C2×C3 | C22×C3×D3 |
| Fitting | Q8×H(3,𝔽3) | C2 |
| hypercenter | Q8 | (C3⋊D3)⋊C3 |
| hypocenter | C32 | C3×#41 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,3,4,7)(2,5,6,8)(9,10,16)(11,13,15)(12,14,17);(1,2,4,6)(3,8,7,5);(3,7)(5,8)(10,12)(11,14)(13,15)(16,17)Polycyclic presentation:
2^3,3,2,3^2|b^2=e,c^2=e,c^a=ce,f^a=f^2,g^a=g^2,c^b=ce,f^d=fg50 conjugacy classes — view the full character table.
| maximal subgroup of | #11721233, #11721235, #11721237, … |
| maximal normal subgroup of | #11721233, #11721235, #11721237, … |
| maximal quotient of | #64099⋊C3, #64101⋊C3, #64102⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 369);