Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C2×C4 |
| direct product | C13×#13285 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C2×D8 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C22×C13 | C2×#102 |
| prime radical | C23×C13 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C2×C42×C13 ◁ C2×C4×C8×C13 ◁ G | |
350 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)(16,20)(17,18)(19,21)(22,23,28,33,29,34,27,44)(24,35,39,51,40,52,38,47)(25,32,26,43,42,46,41,31)(30,45,48,53,49,36,50,37);(15,18)(19,20)(22,24,26,37,28,39,42,45,29,40,41,53,27,38,25,36)(23,30,46,52,44,50,43,51,34,49,32,35,33,48,31,47);(14,16,17,21)(15,19,18,20)(22,25,27,41,29,42,28,26)(23,31,33,32,34,43,44,46)(24,36,38,53,40,45,39,37)(30,47,48,35,49,51,50,52)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6273);