Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#11997 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C13 | C2×D4 |
| commutator | C2×C4 | C22×C8×C13 |
| Frattini, cosocle | C4×C8 | C23×C13 |
| socle | C22×C13 | C2×C4⋊C8 |
| prime radical | C23×C13 | C4⋊C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C13 ◁ C2×C4×C8×C13 ◁ G | |
242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(15,18)(16,20)(19,21)(22,23,25,29,26,30,35,41,27,31,36,42,37,43,47,51)(24,28,33,39,34,40,46,50,32,38,44,48,45,49,52,53);(14,15,16,18,17,19,20,21)(22,24)(23,28)(25,33)(26,34)(27,32)(29,39)(30,40)(31,38)(35,46)(36,44)(37,45)(41,50)(42,48)(43,49)(47,52)(51,53);(24,32)(28,38)(33,44)(34,45)(39,48)(40,49)(46,52)(50,53)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 4985);