Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C4 |
| direct product | C13×#13113 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C2×D8 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C22×C13 | C4×D8 |
| prime radical | C23×C13 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C22×C8×C13 ◁ C2×C4×C8×C13 ◁ G | |
262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 832 |
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)(16,19)(17,18)(20,21)(22,23,28,33,26,31,42,34)(24,32,37,46,36,29,27,43)(25,38,40,50,39,53,41,45)(30,47,49,52,48,44,51,35);(15,18)(19,21)(22,24,28,37,26,36,42,27)(23,29,33,43,31,32,34,46)(25,35,40,47,39,52,41,44)(30,45,49,38,48,50,51,53);(14,16,17,20)(15,19,18,21)(22,25,26,39)(23,30,31,48)(24,35,36,52)(27,44,37,47)(28,40,42,41)(29,45,32,50)(33,49,34,51)(38,46,53,43)754 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6101);