Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C28 |
| direct product | C13×#62572 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C13 |
| prime radical | C13×#3554 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C13×#529 ◁ G | |
1652 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)(16,23)(17,21)(18,27)(19,22)(20,25)(24,29)(26,28)(30,31)(32,41)(33,45)(34,39)(35,40)(36,50)(37,44)(38,46)(42,57)(43,53)(47,54)(48,52)(49,55)(51,58)(56,60)(59,61);(14,16)(15,20)(17,24)(18,26)(19,25)(21,28)(22,23)(27,29)(30,32)(31,37)(33,42)(34,48)(35,44)(36,43)(38,51)(39,53)(40,41)(45,61)(46,60)(47,59)(49,58)(50,52)(54,57)(55,56);(14,17,19,18)(15,21,22,27)(16,24,25,26)(20,28,23,29)(30,33,34,49,35,47,36,46)(31,38,50,54,40,55,39,45)(32,42,48,58,44,59,43,60)(37,51,52,57,41,56,53,61);(16,25)(20,23)(24,26)(28,29)(30,34,35,36)(31,39,40,50)(32,43,44,48)(33,46,47,49)(37,52,41,53)(38,54,55,45)(42,58,59,60)(51,61,56,57);(14,18,19,17)(15,21,22,27)(16,26,25,24)(20,28,23,29)(31,40)(32,44)(38,55)(39,50)(42,59)(43,48)(45,54)(58,60)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 55560);