Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #1088 |
| automorphism | Aut(G) = D4×AΓL(1,𝔽9) (order 1152) · Inn(G) = #157 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived, Sylow | H(3,𝔽3) ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | #157 |
| commutator, hypocenter | H(3,𝔽3) | C2×C8 |
| Frattini, socle | C2×C3 | C2×#157 |
| Fitting | C4×H(3,𝔽3) | C4 |
| prime radical | C2×#272 | C4 |
| cosocle | #1084 | C22 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C3×C4 | |
65 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,3,16)(2,15,6)(4,12,9)(5,10,18)(7,17,11)(8,13,14)(19,21,23,22)(20,24,26,25);(1,2)(3,7,9,14)(4,11)(5,8)(6,12,13,18)(10,15,16,17)(19,20,22,25,23,26,21,24)Polycyclic presentation:
2^4,3^3|a^2=c,b^2=d,c^2=d,e^a=ef^2,f^a=e^2f^2,e^c=e^2,f^c=f^2g,f^e=fg56 conjugacy classes — view the full character table.
| maximal subgroup of | #11720697, (((C3⋊D3)⋊C3)⋊C2)⋊C8, #11720708, … |
| maximal normal subgroup of | #11720697, (((C3⋊D3)⋊C3)⋊C2)⋊C8, #11720708, … |
| maximal quotient of | #11720770, #11720771, C4×#6188, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 273);