Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of #4174 |
| automorphism | Aut(G) = C2×C3×((C3⋊D3)⋊C3)⋊C2 (order 648) · Inn(G) = (C3⋊D3)⋊C3 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived | H(3,𝔽3) ⋊ C3×C4 |
| Sylow | AΣL(1,𝔽27) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2×C3 | (C3⋊D3)⋊C3 |
| commutator, hypocenter | H(3,𝔽3) | C3×C4 |
| Frattini | C2×C32 | C3×D3 |
| Fitting, prime radical | C2×AΣL(1,𝔽27) | C2 |
| cosocle | C2×H(3,𝔽3) | C2×C3 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 13 points:
(2,8,7,5,6,9)(10,11,12,13);(1,2,5)(3,7,9)(4,6,8)Polycyclic presentation:
2,3,2,3^3|a^2=c,d^a=d^2,e^a=e^2f,d^b=de,e^d=ef44 conjugacy classes — view the full character table.
| maximal subgroup of | #4183⋊C4, #4175⋊C4, #10619551, … |
| maximal normal subgroup of | #4183⋊C4, #4175⋊C4, #10619551, … |
| maximal quotient of | #10619500, C2×#66321, #11757282, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 13);