Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C28 |
| direct product | C13×#61877 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C13 |
| prime radical | C2×C13×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C13×#533 ◁ G | |
1600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 61 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 5 generators on 61 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,22)(17,20)(18,21)(19,24)(23,28)(25,27)(26,29)(30,31,36,39)(32,37,42,53)(33,44,46,55)(34,40,35,48)(38,49,54,45)(41,56,57,61)(43,51,47,52)(50,58,60,59);(14,16,18,24)(15,19,21,22)(17,25,26,23)(20,28,29,27)(30,32)(31,37)(33,41)(34,47)(35,43)(36,42)(38,50)(39,53)(40,52)(44,56)(45,59)(46,57)(48,51)(49,58)(54,60)(55,61);(30,33)(31,38)(32,41)(34,45)(35,49)(36,46)(37,50)(39,54)(40,55)(42,57)(43,58)(44,48)(47,59)(51,56)(52,61)(53,60);(14,17,18,26)(15,20,21,29)(16,23,24,25)(19,27,22,28)(31,39)(32,42)(38,54)(40,48)(41,57)(43,47)(44,55)(58,59);(15,21)(19,22)(20,29)(27,28)(30,34,36,35)(31,40,39,48)(32,43,42,47)(33,45,46,49)(37,51,53,52)(38,55,54,44)(41,58,57,59)(50,56,60,61)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 54865);