Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of C3⋊(C3⋊(C3⋊D3)) |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C36 |
| unambiguous semidirect product | C3⋊(C3⋊(C3⋊Dic3)) |
| derived, Sylow | C34 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊(C3⋊(C3⋊D3)) |
| commutator, hypocenter | C34 | C4 |
| Fitting, socle, prime radical, cosocle | C2×C34 | C2 |
| LCS | G ▷ C34 | |
| UCS | C1 ◁ C2 | |
636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 324 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2,3,4)(6,7)(9,10)(12,13)(15,16);(14,15,16);(11,12,13);(8,9,10);(5,6,7)Polycyclic presentation:
2^2,3^4|a^2=b,c^a=c^2,d^a=d^2,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, 158);