Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C28 |
| direct product | C13×#61375 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C13 |
| prime radical | C2×C13×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C13×#533 ◁ G | |
1706 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)(14,15,18,17)(16,20,21,19)(22,23,27,32)(24,33,35,29)(25,37,39,46)(26,31,28,41)(30,42,47,38)(34,49,50,53)(36,45,40,44)(43,52,48,51);(14,16,18,21)(15,19,17,20)(22,24,27,35)(23,29,32,33)(25,34,39,50)(26,40,28,36)(30,43,47,48)(31,45,41,44)(37,53,46,49)(38,52,42,51);(22,25)(23,30)(24,34)(26,38)(27,39)(28,42)(29,43)(31,46)(32,47)(33,48)(35,50)(36,51)(37,41)(40,52)(44,53)(45,49);(14,17,18,15)(16,19,21,20)(24,35)(29,33)(34,50)(36,40)(43,48)(44,45)(49,53)(51,52);(22,26,27,28)(23,31,32,41)(24,36,35,40)(25,38,39,42)(29,44,33,45)(30,46,47,37)(34,51,50,52)(43,53,48,49)754 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 54363);