Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C3 |
No aliases for this group.
| Fitting | C22×C33 ⋊ C22 |
| Sylow | C33 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×C3⋊D3 |
| commutator, socle, cosocle | C2×C33 | C23 |
| Fitting | C22×C33 | C22 |
| hypercenter | C22 | D3×C3⋊D3 |
| hypocenter | C33 | C2×D4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C22 | |
452 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 3 generators on 13 points:
(1,2)(3,4)(8,9,10)(11,12,13);(2,4)(6,7)(9,10);(2,4)(5,6,7)(12,13)Polycyclic presentation:
2^4,3^3|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 | #11724411, #11724419, #11724420, … |
| maximal normal subgroup of | #11724411, #11724419, #11724420, … |
| maximal quotient of | #11723395, #11723420, #11723421, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 686);