Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of C9⋊D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C9 |
| unambiguous semidirect product | C9⋊Dic9 |
| derived, Sylow | C92 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C9⋊D9 |
| commutator, hypocenter | C92 | C4 |
| Frattini, socle, prime radical | C2×C32 | C3⋊D3 |
| Fitting, cosocle | C2×C92 | C2 |
| LCS | G ▷ C92 | |
| UCS | C1 ◁ C2 | |
69 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 324 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3,4)(6,8)(7,11)(9,10)(12,13)(15,17)(16,20)(18,19)(21,22);(14,22,17,20,19,18,16,15,21);(5,13,8,11,10,9,7,6,12)Polycyclic presentation:
2^2,3^4|a^2=b,c^3=e^2,d^3=f^2,c^a=c^2e,d^a=d^2f,e^a=e^2,f^a=f^284 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 19);