Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #88941 |
| automorphism | Aut(G) = C22×D3×GO+(4,𝔽2) (order 1728) · Inn(G) = #157⋊D3 · Out(G) = C23 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C33 ⋊ SD16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #157⋊D3 |
| commutator, cosocle | C3×C3⋊Dic3 | C22 |
| Fitting, socle | C2×C33 | D4 |
| prime radical | #6295 | C2 |
| hypocenter | C33 | SD16 |
| LCS | G ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,3)(4,8,6,5)(7,9)(10,11,12,14,13,15,16,17);(2,3)(7,8)(11,14)(12,16)(15,17)Polycyclic presentation:
2^4,3^3|b^2=c,c^2=d,b^a=bc,c^a=cd,e^a=f,f^a=e,g^a=g^2,e^b=f^2,f^b=e,e^c=e^2,f^c=f^227 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3775, #11724129, #11724133, … |
| maximal normal subgroup of | D3×#3775, #11724129, #11724133, … |
| maximal quotient of | #11722793, #11722794, #11722796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 589);