Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#9671 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | #121 |
| commutator | C22×C4 | C22×C4×C13 |
| Frattini, cosocle | C2×C42 | C23×C13 |
| socle | C22×C13 | #390 |
| prime radical | C23×C13 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C2×C42×C13 ◁ G | |
434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,19,20,18,16,17,21)(22,23)(24,28,33,42)(25,35)(26,38)(27,30)(29,36)(31,39)(32,44,45,53)(34,48,47,41)(37,49)(40,51,52,46)(43,50);(14,16,19,21,18,15,17,20)(22,24,27,33)(23,28)(25,32,37,45)(26,34,39,47)(29,40)(30,42)(31,41)(35,44)(36,46,50,51)(38,48)(43,52)(49,53);(14,17,18,19)(15,20,16,21)(22,25,26,36,27,37,39,50)(23,29,31,35,30,43,38,49)(24,32,34,46,33,45,47,51)(28,40,41,44,42,52,48,53)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 2659);