Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C28 |
| direct product | C13×#60696 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C13 |
| socle | C22×C13 | C23×D4 |
| prime radical | C13×#3061 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C4×C13×#41 ◁ G | |
1850 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)(14,15)(16,19)(17,18)(20,21)(22,23,27,31)(24,28,33,41)(25,35,38,50)(26,39,36,30)(29,43,44,37)(32,45,47,53)(34,49,46,42)(40,51,52,48);(15,18)(19,21)(22,24)(23,28)(25,32)(26,34)(27,33)(29,40)(30,42)(31,41)(35,45)(36,46)(37,48)(38,47)(39,49)(43,51)(44,52)(50,53);(22,25,36,43,27,38,26,37)(23,29,39,50,31,44,30,35)(24,32,46,51,33,47,34,48)(28,40,49,53,41,52,42,45);(14,16,17,20)(15,19,18,21);(24,33)(28,41)(32,47)(34,46)(40,52)(42,49)(45,53)(48,51)832 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 53684);