Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C23×C4 |
| direct product | C13×#33701 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C13 |
| socle | C22×C13 | C22×D8 |
| prime radical | C13×#3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C2×C42×C13 ◁ C2×C4×C8×C13 ◁ G | |
590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 4 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)(24,37)(25,30)(26,32)(27,46)(28,48)(29,36)(31,40)(33,44)(34,47)(35,45)(38,50)(39,51)(41,49)(42,53)(43,52);(15,18)(19,21)(22,24,28,39,26,38,27,25,29,40,44,41,45,42,47,43)(23,30,34,50,35,51,33,37,36,52,46,53,32,49,48,31);(14,16,17,20)(15,19,18,21)(22,25,27,42,45,39,28,40,29,43,47,38,26,41,44,24)(23,31,33,51,32,53,34,30,36,37,48,49,35,50,46,52);(22,26,29,45)(23,32,36,35)(24,38,40,42)(25,41,43,39)(27,44,47,28)(30,49,52,51)(31,53,37,50)(33,34,48,46)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 26689);