Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Sylow | C33 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×C3⋊D3 |
| commutator, socle, cosocle | C2×C33 | C23 |
| Fitting | C33×C4 | C22 |
| hypercenter | C4 | D3×C3⋊D3 |
| hypocenter | C33 | #41 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 | |
288 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 68 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:
(1,4,5,3)(2,6,7,8)(12,13,14)(15,16,17);(1,3,5,4)(2,6,7,8)(10,11)(13,14);(1,2)(3,6)(4,8)(5,7)(9,10,11)(16,17)Polycyclic presentation:
2^4,3^3|b^2=d,c^2=d,c^a=cd,g^a=g^2,e^b=e^2,f^b=f^254 conjugacy classes — view the full character table.
| maximal subgroup of | #11722710, #11722711, #11722857, … |
| maximal normal subgroup of | #11722710, #11722711, #11722857, … |
| maximal quotient of | #11723404, #11723405, #11723406, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 661);