Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×C32×#67 |
| Fitting | C23×C33 ⋊ C2 |
| Sylow | C33 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C32 | C2×D3 |
| commutator, cosocle | C2×C3 | C23×C32 |
| Frattini | C2 | C22×C32×D3 |
| Fitting | C23×C33 | C2 |
| socle | C22×C33 | C22 |
| hypercenter | C23×C32 | D3 |
| hypocenter | C3 | C2×C32×D4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C32 ◁ C23×C32 | |
452 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 15 points:
(3,4,5)(9,10,11)(12,13)(14,15);(6,7,8)(10,11)(13,15);(1,2)162 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#63580, #11722062, C32×#806, … |
| maximal normal subgroup of | C3×#63580, #11722062, C32×#806, … |
| maximal quotient of | C2×C32×Dic3⋊C4, C32×#924, C2×C32×(C2×D3)⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 709);