Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C2×C32 |
No aliases for this group.
| Sylow | C22 ⋊ #737 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C32×A4 |
| commutator, socle, cosocle | C22×C3 | C33 |
| Fitting, prime radical | C22×H(3,𝔽3) | C3 |
| hypercenter | H(3,𝔽3) | A4 |
| hypocenter | C22 | #737 |
| LCS | G ▷ C22×C3 ▷ C22 | |
| UCS | C1 ◁ C3 ◁ H(3,𝔽3) | |
77 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,4)(2,3)(7,13,14)(10,19,20)(12,22,23)(16,25,26)(18,28,29)(24,30,31);(5,7,12)(6,10,18)(8,14,22)(9,16,24)(11,20,28)(13,23,15)(17,26,30)(19,29,21)(25,31,27);(2,3,4)(5,6,9,8,11,17,15,21,27)(7,10,16,14,20,26,13,19,25)(12,18,24,22,28,30,23,29,31)Polycyclic presentation:
3^4,2^2|a^3=d,e^a=f,f^a=ef,c^b=cd44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 129);