Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C28 |
| direct product | C13×#62090 |
| 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)(22,23,27,32)(24,34,37,29)(25,39,41,47)(26,33,28,31)(30,42,46,40)(35,49,51,53)(36,44,38,45)(43,50,48,52);(14,15)(16,19)(17,18)(20,21)(22,24,27,37)(23,29,32,34)(25,35,41,51)(26,36,28,38)(30,43,46,48)(31,44,33,45)(39,53,47,49)(40,50,42,52);(22,25)(23,30)(24,35)(26,40)(27,41)(28,42)(29,43)(31,39)(32,46)(33,47)(34,48)(36,50)(37,51)(38,52)(44,53)(45,49);(15,18)(19,21)(22,26,27,28)(23,31,32,33)(24,36,37,38)(25,40,41,42)(29,44,34,45)(30,39,46,47)(35,50,51,52)(43,53,48,49);(14,16,17,20)(15,19,18,21)(23,32)(24,37)(30,46)(31,33)(35,51)(36,38)(39,47)(50,52)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 55078);