Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C28 |
| direct product | C13×#62168 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C13 |
| prime radical | C13×#121 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C13×#529 ◁ G | |
1628 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)(14,15,18,21)(16,22,23,19)(17,25,26,20)(24,27,29,28)(30,31)(32,40)(33,44)(34,38)(35,48)(36,42)(37,46)(39,47)(41,56)(43,51)(45,53)(49,52)(50,60)(54,58)(55,59)(57,61);(14,16)(15,19)(17,24)(18,23)(20,27)(21,22)(25,28)(26,29)(30,32)(31,36)(33,41)(34,42)(35,43)(37,50)(38,40)(39,51)(44,61)(45,57)(46,58)(47,59)(48,55)(49,60)(52,54)(53,56);(30,33,34,45)(31,37,38,52)(32,41,42,57)(35,49,47,46)(36,50,40,54)(39,53,48,44)(43,60,59,58)(51,56,55,61);(16,23)(19,22)(24,29)(27,28)(32,42)(36,40)(41,57)(43,59)(50,54)(51,55)(56,61)(58,60);(14,17,18,26)(15,20,21,25)(16,24,23,29)(19,27,22,28)(31,38)(32,42)(37,52)(39,48)(41,57)(43,59)(44,53)(58,60)637 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 55156);