Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of (C3⋊#226)⋊C3 |
| automorphism | Aut(G) = C2×((C3⋊S4)⋊C3)⋊C2 (order 864) · Inn(G) = (C3⋊S4)⋊C3 · Out(G) = C22 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | (C3⋊#226)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | (C3⋊S4)⋊C3 |
| commutator, cosocle, hypocenter | C3×SL(2,𝔽3) | C2×C3 |
| Frattini, socle | C2×C3 | C3×S4 |
| Fitting | C32×Q8 | D3 |
| prime radical | Q8⋊H(3,𝔽3) | C2 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,6,10,13,5,15,12,7)(2,11,14,4,8,3,16,9)(18,23,25,20,21,24)(19,22);(3,9,12)(4,10,11)(6,13,16)(7,14,15)(17,18,20)(19,21,24)(22,25,23)Polycyclic presentation:
2,3^3,2^3|a^2=g,e^2=g,f^2=g,c^a=c^2,d^a=d^2,e^a=f,f^a=e,c^b=cd,e^c=fg,f^c=ef,f^e=fg29 conjugacy classes — view the full character table.
| maximal subgroup of | #11722253, #11722254, #11722256, … |
| maximal normal subgroup of | #11722253, #11722254, #11722256, … |
| maximal quotient of | (C3⋊#859)⋊C3, C2×(C3⋊#226)⋊C3, #13046952, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 247);