Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C28 |
| direct product | C13×#62565 |
| 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 | |
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)(30,31,35,39)(32,36,41,49)(33,43,44,58)(34,46,47,38)(37,51,52,45)(40,53,54,61)(42,57,56,50)(48,59,60,55);(14,16)(15,20)(17,24)(18,26)(19,25)(21,28)(22,23)(27,29)(30,32)(31,36)(33,40)(34,42)(35,41)(37,48)(38,50)(39,49)(43,53)(44,54)(45,55)(46,57)(47,56)(51,59)(52,60)(58,61);(14,17,19,18)(15,21,22,27)(16,24,25,26)(20,28,23,29)(30,33,34,45,35,44,47,51)(31,37,38,43,39,52,46,58)(32,40,42,55,41,54,56,59)(36,48,50,53,49,60,57,61);(16,25)(20,23)(24,26)(28,29)(33,44)(37,52)(40,54)(43,58)(45,51)(48,60)(53,61)(55,59);(14,18,19,17)(15,21,22,27)(16,26,25,24)(20,28,23,29)(32,41)(36,49)(40,54)(42,56)(48,60)(50,57)(53,61)(55,59)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 55553);