Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of #4174⋊C2 |
| automorphism | Aut(G) = (#4174⋊C2)⋊C3 (order 972) · Inn(G) = G · Out(G) = C3 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #4174⋊C2 |
| unambiguous semidirect product | #4175⋊C2 |
| unambiguous semidirect product | #4183⋊C2 |
| Fitting, derived, Sylow | AΣL(1,𝔽27) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | AΣL(1,𝔽27) | C22 |
| Frattini | C32 | D32 |
| socle | C3 | ((C3⋊D3)⋊C3)⋊C2 |
| LCS | G ▷ AΣL(1,𝔽27) | |
| UCS | C1 | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 9 |
| transitive degree | μt(G) = 9 |
A minimal faithful permutation representation — 2 generators on 9 points:
(1,2,5)(3,6,9,4,7,8);(2,8,7,5,6,9)Polycyclic presentation:
2^2,3^4|d^a=d^2,e^a=e^2f,c^b=c^2,e^b=e^2,f^b=f^2,d^c=de,e^d=ef17 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4174⋊C2, C3×#4174⋊C2, (#4174⋊C2)⋊C3, … |
| maximal normal subgroup of | C2×#4174⋊C2, C3×#4174⋊C2, (#4174⋊C2)⋊C3, … |
| maximal quotient of | #4174⋊C4, #4183⋊C4, #4175⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 39);