Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #88436 |
| automorphism | Aut(G) = C2×D4×((C3⋊D3)⋊C3)⋊C2 (order 1728) · Inn(G) = #6200 · Out(G) = C23 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | H(3,𝔽3) ⋊ D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #6200 |
| commutator, Fitting, cosocle | C4×H(3,𝔽3) | C22 |
| Frattini | C3×C4 | D32 |
| socle | C2×C3 | #626 |
| hypercenter | C4 | ((C3⋊D3)⋊C3)⋊C2 |
| hypocenter | H(3,𝔽3) | D8 |
| LCS | G ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 | |
93 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 17 points:
(2,8,6,4,5,9)(3,7)(10,11,15,17,13,16,12,14);(1,2,4)(3,5,8,7,6,9)(11,14)(12,15)(16,17)Polycyclic presentation:
2^4,3^3|b^2=cd,c^2=d,b^a=bc,c^a=cd,f^a=f^2,g^a=g^2,e^b=e^2,g^b=g^2,f^e=fg29 conjugacy classes — view the full character table.
| maximal subgroup of | #11720847, #11720848, #11720849, … |
| maximal normal subgroup of | #11720847, #11720848, #11720849, … |
| maximal quotient of | #11720253, #11720254, #11720255, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 83);