Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of #4183 |
| automorphism | Aut(G) = C2×(#4174⋊C2)⋊C3 (order 1944) · Inn(G) = #4183 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | AΣL(1,𝔽27) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #4183 |
| commutator, hypocenter | AΣL(1,𝔽27) | C4 |
| Frattini | C2×C32 | C3⋊D3 |
| Fitting, prime radical, cosocle | C2×AΣL(1,𝔽27) | C2 |
| socle | C2×C3 | #272 |
| LCS | G ▷ AΣL(1,𝔽27) | |
| UCS | C1 ◁ C2 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 3 generators on 13 points:
(1,2,3,4)(6,9)(7,8)(10,13)(11,12);(6,10,11)(9,12,13);(5,6,9)(7,10,12)(8,11,13)Polycyclic presentation:
2^2,3^4|a^2=b,c^a=c^2,d^a=d^2,e^a=ef^2,f^a=f^2,d^c=de,e^c=ef26 conjugacy classes — view the full character table.
| maximal subgroup of | #4174⋊C4, #4175⋊C4, #10619550, … |
| maximal normal subgroup of | #4174⋊C4, #4175⋊C4, #10619550, … |
| maximal quotient of | #10619509, C2×#66330, #11757321, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 22);