Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C28 |
| direct product | C13×#61422 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C13 |
| prime radical | C13×#3035 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C13×#533 ◁ G | |
1844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 5 generators on 53 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(15,19)(20,21)(22,23,27,30)(24,28,33,43)(25,35,37,45)(26,38,40,31)(29,39,44,36)(32,46,47,53)(34,50,49,42)(41,51,52,48);(14,15,18,19)(16,20,17,21)(22,24)(23,28)(25,32)(26,34)(27,33)(29,41)(30,43)(31,42)(35,46)(36,48)(37,47)(38,50)(39,51)(40,49)(44,52)(45,53);(14,16,18,17)(15,20,19,21)(22,25,27,37)(23,29,30,44)(24,32,33,47)(26,39,40,36)(28,41,43,52)(31,45,38,35)(34,51,49,48)(42,53,50,46);(14,17,18,16)(15,20,19,21);(23,30)(24,33)(29,44)(31,38)(32,47)(34,49)(35,45)(48,51)715 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 54410);