Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#32793 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C13 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C13 |
| prime radical | C24×C13 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C2×C13×#117 ◁ G | |
592 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 4 generators on 53 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,18)(17,19)(20,21)(22,23,27,30)(24,31,25,28)(26,29,34,37)(32,36,33,35)(38,39,42,44)(40,43,48,51)(41,45,49,53)(46,50,47,52);(14,16)(15,18)(17,20)(19,21)(22,24,27,25)(23,28,30,31)(26,33,34,32)(29,36,37,35)(38,40)(39,43)(41,46)(42,48)(44,51)(45,50)(47,49)(52,53);(15,19)(18,21)(22,25,27,24)(23,28,30,31)(26,32,34,33)(29,36,37,35)(40,46)(41,49)(43,50)(45,53)(47,48)(51,52);(16,20)(18,21)(22,26,27,34)(23,29,30,37)(24,32,25,33)(28,35,31,36)(39,44)(43,51)(45,53)(50,52)442 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 25781);