Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of #1088 |
| automorphism | Aut(G) = C3×AΓL(1,𝔽9) (order 432) · Inn(G) = #157 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, Sylow | #737 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | #157 |
| commutator, hypocenter | H(3,𝔽3) | C3×C4 |
| Frattini, socle | C3 | C3×#157 |
| Fitting | #737 | C4 |
| prime radical, cosocle | #272 | C2×C3 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C9 | |
31 conjugacy classes of subgroups.
| minimal degree | μ(G) = 54 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 2 generators on 54 points:
(1,31,28,10,3,44,5,22,11,2,13,38,6,7,27,17,14,19)(4,39,52,34,12,49,35,47,26,9,48,53,16,21,54,8,30,37,15,20,43,25,29,36,18,41,42,24,32,51,33,40,45,23,46,50);(1,4,33)(2,25,8)(3,12,46)(5,18,15)(6,16,35)(7,41,20)(9,34,17)(10,23,24)(11,26,52)(13,32,29)(14,30,48)(19,51,36)(21,47,31)(22,39,40)(27,45,42)(28,43,54)(37,53,44)(38,49,50)Polycyclic presentation:
2,3,2,3^3|a^2=c,b^3=f,d^a=de^2,e^a=d^2e^2,d^c=d^2,e^c=e^2f,e^d=ef42 conjugacy classes — view the full character table.
| derived subgroup of | #10620020⋊C2 |
| maximal subgroup of | #10620020, #66419⋊C2, #10620032, … |
| maximal normal subgroup of | #10620020, #66419⋊C2, #10620032, … |
| maximal quotient of | #10619833, C2×#66419, C9×#1088, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 111);