Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C23×C4 |
| direct product | C13×#33697 |
| 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 | |
598 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)(24,31)(25,30)(26,32)(27,34)(28,33)(29,36)(35,43)(37,49)(38,45)(39,50)(40,46)(41,48)(42,51)(44,52)(47,53);(15,18)(19,21)(22,24,28,38,26,37,42,41,29,39,44,53,43,40,27,25)(23,30,34,46,35,47,52,50,36,48,51,49,32,45,33,31);(14,16,17,20)(15,19,18,21)(22,25,27,40,43,53,44,39,29,41,42,37,26,38,28,24)(23,31,33,45,32,49,51,48,36,50,52,47,35,46,34,30);(22,26,29,43)(23,32,36,35)(24,37,39,40)(25,38,41,53)(27,28,42,44)(30,45,48,47)(31,49,50,46)(33,51,52,34)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 26685);