Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C23×C4 |
| direct product | C13×#33708 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C13 |
| prime radical | C13×#3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C2×C42×C13 ◁ C2×C4×C8×C13 ◁ G | |
590 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)(16,19)(17,18)(20,21)(22,23)(24,37)(25,30)(26,32)(27,46)(28,48)(29,36)(31,42)(33,38)(34,47)(35,45)(39,53)(40,49)(41,50)(43,52)(44,51);(15,18)(19,21)(22,24,38,43,26,41,47,44,29,42,28,40,45,39,27,25)(23,30,46,53,35,49,48,31,36,51,34,50,32,52,33,37);(14,16,17,20)(15,19,18,21)(22,25,27,39,45,40,28,42,29,44,47,41,26,43,38,24)(23,31,33,49,32,53,34,30,36,37,48,52,35,50,46,51);(22,26,29,45)(23,32,36,35)(24,39,42,41)(25,40,44,43)(27,38,47,28)(30,49,51,52)(31,50,37,53)(33,34,48,46)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 26696);