Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C28 |
| direct product | C13×#62080 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C13 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C13×#529 ◁ G | |
1778 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)(15,18)(16,20)(22,23,28,33)(24,29,36,44)(25,37,39,47)(26,32,27,31)(30,41,45,38)(34,48,50,52)(35,46,40,43)(42,49,53,51);(14,15)(16,19)(17,18)(20,21)(22,24)(23,29)(25,34)(26,40)(27,35)(28,36)(30,42)(31,46)(32,43)(33,44)(37,48)(38,51)(39,50)(41,49)(45,53)(47,52);(22,25)(23,30)(24,34)(26,38)(27,41)(28,39)(29,42)(31,37)(32,47)(33,45)(35,49)(36,50)(40,51)(43,52)(44,53)(46,48);(15,18)(19,21)(22,26,28,27)(23,31,33,32)(24,35,36,40)(25,38,39,41)(29,43,44,46)(30,37,45,47)(34,49,50,51)(42,52,53,48);(14,16,17,20)(15,19,18,21)(24,36)(29,44)(34,50)(35,40)(42,53)(43,46)(48,52)(49,51)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 55068);