Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C28 |
| direct product | C13×#61693 |
| 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×#532 ◁ G | |
2014 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)(22,23,27,33)(24,31,37,29)(25,38,40,48)(26,34,28,32)(30,42,46,39)(35,49,51,53)(36,44,41,47)(43,52,45,50);(14,16)(15,18)(17,21)(19,20)(22,24)(23,29)(25,35)(26,41)(27,37)(28,36)(30,43)(31,33)(32,47)(34,44)(38,53)(39,52)(40,51)(42,50)(45,46)(48,49);(22,25,27,40)(23,30,33,46)(24,35,37,51)(26,39,28,42)(29,43,31,45)(32,38,34,48)(36,50,41,52)(44,49,47,53);(16,21)(18,20)(22,24)(23,31)(25,35)(26,41)(27,37)(28,36)(29,33)(30,45)(32,44)(34,47)(38,49)(39,52)(40,51)(42,50)(43,46)(48,53);(15,19)(18,20)(22,26,27,28)(23,32,33,34)(24,36,37,41)(25,39,40,42)(29,44,31,47)(30,38,46,48)(35,50,51,52)(43,49,45,53)637 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 54681);