Finite Group DatabasebetaFinite Group Databasebeta| order | 333 |
| isoclinism | isoclinic class of #66526 |
| automorphism | Aut(G) = AGL(1,𝔽37) (order 1332) · Inn(G) = G · Out(G) = C4 |
| homology | Abelianization (H1) = C9 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Sylow | C37 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C37 | C9 |
| prime radical, cosocle | C37⋊C3 | C3 |
| LCS | G ▷ C37 | |
| UCS | C1 | |
6 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 37 |
A minimal faithful permutation representation — 2 generators on 37 points:
(2,3,6,4,7,14,9,16,26)(5,11,23,12,18,31,24,32,30)(8,17,29,15,27,37,28,33,36)(10,20,34,21,35,13,19,22,25);(1,2,5,13,25,27,37,26,36,7,9,19,16,17,30,12,3,8,18,33,24,35,28,29,32,20,4,10,22,31,34,23,11,14,6,15,21)Polycyclic presentation:
3^2,37|a^3=b,c^a=c^16,c^b=c^2613 conjugacy classes — view the full character table.
| maximal subgroup of | #66526⋊C2, C2×#66526, C3×#66526, … |
| maximal normal subgroup of | #66526⋊C2, C2×#66526, C3×#66526, … |
| maximal quotient of | C2×#66526, #11758603, C3×#66526, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(333, 3);