Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#12000 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | #106 |
| commutator | C8 | C22×C8×C13 |
| Frattini, cosocle | C4×C8 | C23×C13 |
| socle | C22×C13 | C2×#106 |
| prime radical | C23×C13 | #106 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C2×C42×C13 ◁ C2×C4×C8×C13 ◁ G | |
242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,18,17,19,20,21)(23,27,28,33)(24,30)(25,32)(29,36,35,34)(31,37);(22,23,24,27,25,29,30,34,26,28,31,33,32,35,37,36);(23,28)(27,33)(29,35)(34,36)832 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 4988);