Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of C3⋊#3688 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊#3688 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊((C22⋊C9)⋊C2) |
| commutator, cosocle, hypocenter | C3×Q8⋊C9 | C2 |
| Frattini | C2×C3 | C3⋊S4 |
| Fitting | C32×Q8 | D3 |
| socle, prime radical | C2×C32 | S4 |
| LCS | G ▷ C3×Q8⋊C9 | |
| UCS | C1 ◁ C2 | |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,6,10,13,5,15,12,7)(2,11,14,4,8,3,16,9)(18,19)(21,23)(22,26)(24,25)(27,28);(17,18,19);(3,9,12)(4,10,11)(6,13,16)(7,14,15)(20,28,23,26,25,24,22,21,27)Polycyclic presentation:
2,3^3,2^3|a^2=g,b^3=d^2,e^2=g,f^2=g,b^a=b^2d,c^a=c^2,d^a=d^2,e^a=f,f^a=e,e^b=fg,f^b=ef,f^e=fg36 conjugacy classes — view the full character table.
| maximal subgroup of | #11722245, #11722246, D3×#3688, … |
| maximal normal subgroup of | #11722245, #11722246, D3×#3688, … |
| maximal quotient of | C3⋊#63421, C2×C3⋊#3688, C3⋊#88390, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 255);