Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C28 |
| direct product | C13×#61413 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C13 |
| prime radical | C13×#3155 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C13×#533 ◁ G | |
1886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 832 |
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)(22,23,27,32)(24,33,35,29)(25,37,39,46)(26,31,28,41)(30,42,47,38)(34,49,50,53)(36,45,40,44)(43,52,48,51);(14,15,18,19)(16,20,17,21)(22,24,27,35)(23,29,32,33)(25,34,39,50)(26,40,28,36)(30,43,47,48)(31,45,41,44)(37,53,46,49)(38,52,42,51);(14,16,18,17)(15,20,19,21)(22,25,27,39)(23,30,32,47)(24,34,35,50)(26,38,28,42)(29,43,33,48)(31,46,41,37)(36,51,40,52)(44,53,45,49);(14,17,18,16)(15,20,19,21)(24,35)(29,33)(34,50)(36,40)(43,48)(44,45)(49,53)(51,52);(22,26,27,28)(23,31,32,41)(24,36,35,40)(25,38,39,42)(29,44,33,45)(30,46,47,37)(34,51,50,52)(43,53,48,49)754 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 54401);