Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of #4181 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C32⋊C9 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C3⋊D9 |
| commutator, hypocenter | C32⋊C9 | C4 |
| Frattini, socle | C2×C32 | C3⋊D3 |
| Fitting, cosocle | C2×C32⋊C9 | C2 |
| prime radical | C2×C33 | D3 |
| LCS | G ▷ C32⋊C9 | |
| UCS | C1 ◁ C2×C3 | |
61 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 3 generators on 22 points:
(2,4)(3,7)(5,6)(8,9)(11,13)(14,18)(15,17)(19,20,21,22);(11,14,15)(13,17,18);(1,9,4,7,6,5,3,2,8)(10,11,13)(12,15,17)(14,18,16)Polycyclic presentation:
2^2,3^4|a^2=b,c^3=e^2,c^a=c^2e,d^a=d^2,e^a=e^2,d^c=df42 conjugacy classes — view the full character table.
| maximal subgroup of | ((C3⋊D3)⋊C9)⋊C4, #4169⋊C4, #10619536, … |
| maximal normal subgroup of | ((C3⋊D3)⋊C9)⋊C4, #4169⋊C4, #10619536, … |
| maximal quotient of | #10619507, C2×#66328, #11757320, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 20);