Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#11662 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C2×D8 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C4×C8 | C23×C13 |
| socle | C22×C13 | C2×#102 |
| prime radical | C23×C13 | #102 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C2×C42×C13 ◁ C2×C4×C8×C13 ◁ G | |
358 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)(14,15,16,19,17,18,20,21)(22,23)(24,33)(25,30)(26,39)(27,43)(28,32)(29,35)(31,41)(34,47)(36,53)(37,50)(38,48)(40,46)(42,49)(44,51)(45,52);(22,24,28,37)(23,29,32,45)(25,36,27,34)(26,40,42,51)(30,46,43,44)(31,47,48,53)(33,49,50,39)(35,38,52,41);(15,18)(19,21)(22,25,28,27)(23,30,32,43)(24,34,37,36)(26,38,42,41)(29,44,45,46)(31,39,48,49)(33,47,50,53)(35,51,52,40)754 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 4650);