Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C28 |
| direct product | C13×#61552 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C13 |
| prime radical | C13×#3515 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C13×#529 ◁ G | |
1756 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)(15,18)(19,21)(22,23,28,32)(24,29,36,46)(25,38,41,49)(26,33,27,31)(30,42,48,40)(34,50,39,53)(35,45,37,44)(43,52,47,51);(14,15)(16,19)(17,18)(20,21)(22,24)(23,29)(25,39)(26,35)(27,37)(28,36)(30,47)(31,44)(32,46)(33,45)(34,41)(38,53)(40,52)(42,51)(43,48)(49,50);(22,25)(23,30)(24,34)(26,40)(27,42)(28,41)(29,43)(31,38)(32,48)(33,49)(35,51)(36,39)(37,52)(44,50)(45,53)(46,47);(15,18)(19,21)(22,26,28,27)(23,31,32,33)(24,35,36,37)(25,40,41,42)(29,44,46,45)(30,38,48,49)(34,51,39,52)(43,50,47,53);(14,16,17,20)(15,19,18,21)(23,32)(24,36)(30,48)(31,33)(34,39)(35,37)(38,49)(51,52)715 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 54540);