Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C13 · Schur multiplier (H2) = C23×C4 |
| direct product | C13×#21140 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C22×D4 |
| commutator | C2×C4 | C23×C4×C13 |
| Frattini, cosocle | C22×C4 | C24×C13 |
| socle | C22×C13 | C2×C4×D4 |
| prime radical | C2×C13×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C2×C13×#97 ◁ G | |
764 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 832 |
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)(16,19)(17,18)(20,21)(22,23,27,32,28,31,26,40)(24,29,36,43,34,44,35,50)(25,37,38,30,39,46,41,45)(33,47,49,42,48,52,51,53);(15,18)(19,21)(22,24,27,36,28,34,26,35)(23,29,32,43,31,44,40,50)(25,33,38,49,39,48,41,51)(30,42,46,52,45,53,37,47);(22,25)(23,30)(24,33)(26,41)(27,38)(28,39)(29,42)(31,45)(32,46)(34,48)(35,51)(36,49)(37,40)(43,52)(44,53)(47,50);(14,16,17,20)(15,19,18,21)(23,31)(24,34)(30,45)(32,40)(33,48)(35,36)(37,46)(49,51)832 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 14128);