Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of #4169 |
| automorphism | Aut(G) = C2×(((C3⋊D3)⋊C9)⋊C2)⋊C3 (order 1944) · Inn(G) = #4169 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C3 |
No aliases for this group.
| derived | C3×C9 ⋊ C3×C4 |
| Sylow | C32⋊C9 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #4169 |
| commutator, hypocenter | C3×C9 | C3×C4 |
| Frattini, socle | C2×C32 | C3×D3 |
| Fitting | C2×C32⋊C9 | C2 |
| prime radical | C2×C33 | D3 |
| cosocle | C2×C3×C9 | C2×C3 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 ◁ C2 | |
53 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 2 generators on 22 points:
(2,7,9,4,5,8)(3,6)(11,13)(12,16)(14,15)(17,18)(19,20,21,22);(1,2,4)(3,5,8)(6,9,7)(10,18,13,16,15,14,12,11,17)Polycyclic presentation:
2,3,2,3^3|a^2=c,d^3=f^2,d^a=d^2f,e^a=e^2,f^a=f^2,d^b=de42 conjugacy classes — view the full character table.
| maximal subgroup of | ((C3⋊D3)⋊C9)⋊C4, #4181⋊C4, #10619535, … |
| maximal normal subgroup of | ((C3⋊D3)⋊C9)⋊C4, #4181⋊C4, #10619535, … |
| maximal quotient of | #10619495, C2×#66316, #11757281, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 8);