Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Sylow | C33 ⋊ C2×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×C3⋊D3 |
| commutator, socle, cosocle | C2×C33 | C23 |
| Fitting | C33×C4 | C22 |
| prime radical | C2×C3⋊(C3⋊D3) | C22 |
| hypercenter | C4 | D3×C3⋊D3 |
| hypocenter | C33 | C2×Q8 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 | |
292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,4,5,8)(2,6,3,7)(12,13,14)(15,16,17);(1,3,5,2)(4,6,8,7)(10,11)(13,14);(1,2,5,3)(4,7,8,6)(9,10,11)(16,17)Polycyclic presentation:
2^4,3^3|a^2=d,b^2=d,c^2=d,c^a=cd,g^a=g^2,c^b=cd,e^b=e^2,f^b=f^251 conjugacy classes — view the full character table.
| maximal subgroup of | #11722841, #11722848, #11722893, … |
| maximal normal subgroup of | #11722841, #11722848, #11722893, … |
| maximal quotient of | #11723391, #11723393, #11723394, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 666);