Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of A4 |
| automorphism | Aut(G) = S4×C27 (order 648) · Inn(G) = A4 · Out(G) = C2×C27 |
| homology | Abelianization (H1) = C81 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C22⋊C81 |
| derived, Sylow | C22 ⋊ C81 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C27 | A4 |
| commutator, hypocenter | C22 | C81 |
| Fitting, cosocle | C22×C27 | C3 |
| socle, prime radical | C22×C3 | C27 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C27 | |
14 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 162 |
A minimal faithful permutation representation — 2 generators on 85 points:
(2,3,4)(5,6,10,7,11,20,14,23,36,8,12,21,15,24,37,29,42,55,17,26,39,31,44,57,49,62,71,9,13,22,16,25,38,30,43,56,18,27,40,32,45,58,50,63,72,34,47,60,52,65,74,68,77,81,19,28,41,33,46,59,51,64,73,35,48,61,53,66,75,69,78,82,54,67,76,70,79,83,80,84,85);(1,4)(2,3)Polycyclic presentation:
3^4,2^2|a^3=b,b^3=c,c^3=d,e^a=f,f^a=ef108 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 3);