Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C27×C4 |
| direct product | C13×#60821 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C13 |
| prime radical | C13×#3499 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C4×C13×#41 ◁ G | |
2002 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)(16,20)(17,19)(18,21)(22,23,27,31)(24,32,35,45)(25,29,36,43)(26,38,40,30)(28,39,42,34)(33,46,47,53)(37,50,49,44)(41,51,52,48);(14,16,17,21)(15,18,19,20);(16,21)(18,20)(22,24,27,35)(23,28,31,42)(25,33,36,47)(26,39,40,34)(29,41,43,52)(30,45,38,32)(37,51,49,48)(44,53,50,46);(15,19)(18,20)(22,25)(23,29)(24,33)(26,37)(27,36)(28,41)(30,44)(31,43)(32,46)(34,48)(35,47)(38,50)(39,51)(40,49)(42,52)(45,53);(25,36)(29,43)(33,47)(37,49)(41,52)(44,50)(46,53)(48,51)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 53809);