Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of C9⋊A4 |
| automorphism | Aut(G) = A4⋊#4208 (order 1944) · Inn(G) = C3×A4 · Out(G) = C3×C3⋊D3 |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C22 ⋊ C27⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C9 | C3×A4 |
| commutator, socle | C22×C3 | C3×C9 |
| Fitting | C22×C3×C9 | C3 |
| prime radical | C22×C32 | C9 |
| cosocle | C22×C9 | C32 |
| hypercenter | C3×C9 | A4 |
| hypocenter | C22 | C27⋊C3 |
| LCS | G ▷ C22×C3 ▷ C22 | |
| UCS | C1 ◁ C9 ◁ C3×C9 | |
34 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,4)(2,3)(6,10,12)(9,16,18)(11,19,21)(17,24,26)(20,27,28)(25,30,31);(2,3,4)(5,6,9,7,11,17,13,20,25,8,12,16,14,21,24,22,28,30,15,10,18,23,19,26,29,27,31)Polycyclic presentation:
3^4,2^2|a^3=c,c^3=d,b^a=bd,e^a=f,f^a=ef60 conjugacy classes — view the full character table.
| maximal subgroup of | ((C22⋊C27)⋊C2)⋊C3, C2×#66353, #11757417, … |
| maximal normal subgroup of | ((C22⋊C27)⋊C2)⋊C3, C2×#66353, #11757417, … |
| maximal quotient of | #10619570, C2×#66353, #11757394, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 45);