Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| automorphism | Aut(G) = C23×D33 (order 1728) · Inn(G) = D33 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C33 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D33 |
| commutator, Fitting, socle, cosocle | C2×C33 | C23 |
| hypocenter | C33 | #41 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 | |
218 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 218 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 24 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,3,4,7)(2,5,6,8)(10,11)(12,13,14)(15,16,17);(1,2,4,6)(3,8,7,5)(9,10,11)(13,14);(3,7)(5,8)(16,17)Polycyclic presentation:
2^4,3^3|b^2=d,c^2=d,c^a=cd,g^a=g^2,c^b=cd,f^b=f^2,e^c=e^242 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3796, #11724377, D3×#3799, … |
| maximal normal subgroup of | D3×#3796, #11724377, D3×#3799, … |
| maximal quotient of | #11722914, #11722920, #11722922, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 609);