Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C28 |
| direct product | C13×#61350 |
| 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 | |
1746 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)(19,20)(22,23,27,32)(24,34,36,29)(25,38,40,47)(26,33,28,31)(30,42,45,39)(35,49,50,53)(37,44,41,46)(43,52,48,51);(14,15,17,18)(16,20,21,19)(22,24,27,36)(23,29,32,34)(25,35,40,50)(26,41,28,37)(30,43,45,48)(31,46,33,44)(38,53,47,49)(39,52,42,51);(22,25)(23,30)(24,35)(26,39)(27,40)(28,42)(29,43)(31,38)(32,45)(33,47)(34,48)(36,50)(37,51)(41,52)(44,49)(46,53);(14,16,17,21)(15,19,18,20)(24,36)(29,34)(35,50)(37,41)(43,48)(44,46)(49,53)(51,52);(22,26,27,28)(23,31,32,33)(24,37,36,41)(25,39,40,42)(29,44,34,46)(30,38,45,47)(35,51,50,52)(43,49,48,53)715 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 54338);