Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #6290 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22×C3 |
| unambiguous semidirect product | (C2×D3)⋊(C3⋊Dic3) |
| Sylow | C33 ⋊ C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | D3×C3⋊D3 |
| commutator | C2×C33 | C2×C4 |
| Fitting, socle, cosocle | C22×C33 | C22 |
| prime radical | C22×C32×D3 | C2 |
| hypocenter | C33 | C22⋊C4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C22 | |
268 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,3,4)(6,8)(10,11)(13,14)(15,16,17);(5,6)(7,8)(12,13,14)(16,17);(9,10,11)Polycyclic presentation:
2^4,3^3|b^2=c,b^a=bd,g^a=g^2,e^b=e^2,f^b=f^266 conjugacy classes — view the full character table.
| maximal subgroup of | D3×(C2×D3)⋊Dic3, #11722902, #11722913, … |
| maximal normal subgroup of | D3×(C2×D3)⋊Dic3, #11722902, #11722913, … |
| maximal quotient of | (C2×D3)⋊(C3⋊(C3⋊C8)), #11722025, #11722028, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 449);