Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #6308 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C33 |
No aliases for this group.
| Fitting | C2×C33×C4 ⋊ C2 |
| Sylow | C33 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×C3⋊(C3⋊D3) |
| commutator, socle, cosocle | C2×C33 | C23 |
| Frattini | C2 | C22×C3⋊(C3⋊D3) |
| Fitting | C2×C33×C4 | C2 |
| hypercenter | C2×C4 | C3⋊(C3⋊D3) |
| hypocenter | C33 | #41 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
560 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 4 generators on 17 points:
(2,6)(5,8)(15,16,17);(1,3,4,7)(2,5,6,8)(12,13,14);(1,2)(3,5)(4,6)(7,8)(10,11)(13,14)(16,17);(9,10,11)Polycyclic presentation:
2^4,3^3|b^2=d,c^a=cd,e^a=e^2,f^a=f^2,g^a=g^2114 conjugacy classes — view the full character table.
| maximal subgroup of | #11722047, #11722048, #11722175, … |
| maximal normal subgroup of | #11722047, #11722048, #11722175, … |
| maximal quotient of | C4×#6308, #11723911, #11723913, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 723);