Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C24 |
| direct product | C13×#33726 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C13 |
| socle | C22×C13 | C22×D8 |
| prime radical | C23×C13 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C2×C42×C13 ◁ C2×C4×C8×C13 ◁ G | |
490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 4 generators on 53 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,17,19)(16,20,21,18)(22,23)(24,33)(25,30)(26,40)(27,42)(28,32)(29,36)(31,34)(35,46)(37,52)(38,48)(39,45)(41,53)(43,50)(44,51)(47,49);(14,16,17,21)(15,18,19,20);(22,24,34,49,27,37,41,51,28,38,39,50,25,35,26,36)(23,29,40,46,30,43,45,48,32,44,53,52,42,47,31,33);(22,25,28,27)(23,30,32,42)(24,35,38,37)(26,39,41,34)(29,43,44,47)(31,40,45,53)(33,46,48,52)(36,50,51,49)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 26714);