Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of #66431 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting, derived, Sylow | C32×C9 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C32×C9 | C22 |
| Frattini | C3 | #1113 |
| socle | C33 | C2×D3 |
| LCS | G ▷ C32×C9 | |
| UCS | C1 | |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2)(4,5,6)(8,10)(9,13)(11,12)(14,15);(2,3)(5,6)(7,15,10,13,12,11,9,8,14)Polycyclic presentation:
2^2,3^4|d^3=f^2,c^a=c^2,e^a=e^2,c^b=c^2,d^b=d^2f,f^b=f^233 conjugacy classes — view the full character table.
| maximal subgroup of | D32⋊D9, #157⋊D9, D32×D9, … |
| maximal normal subgroup of | D32⋊D9, #157⋊D9, D32×D9, … |
| maximal quotient of | #10619895, #10619896, #10619897, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 123);