Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of 3−1+2⋊Q8 |
| automorphism | Aut(G) = #120×AGL(1,ℤ/9) (order 1728) · Inn(G) = C2×AGL(1,ℤ/9) · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×#611⋊C3 |
| unambiguous semidirect product | (C2×#611)⋊C3 |
| Fitting | C23×3−1+2 ⋊ C2 |
| Sylow | 3−1+2 ⋊ C2×D4 |
178 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 62 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)(5,6)(7,11,12,13,8,14,9,15,10);(4,6)(8,10)(9,13)(11,12)(14,15);(1,2)62 conjugacy classes — view the full character table.
| maximal subgroup of | #63365⋊C3, (#611⋊C3)⋊C4, #63444⋊C3, … |
| maximal normal subgroup of | #63365⋊C3, (#611⋊C3)⋊C4, #63444⋊C3, … |
| maximal quotient of | C2×(3−1+2⋊C4)⋊C4, #63486⋊C3, C2×(C2×AGL(1,ℤ/9))⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 379);