Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C24 |
| direct product | C13×#33735 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C13 |
| socle | C22×C13 | C22×D8 |
| prime radical | C13×#2215 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C2×C42×C13 ◁ C2×C4×C8×C13 ◁ G | |
526 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)(24,29)(25,35)(26,40)(27,42)(28,32)(30,37)(31,36)(33,46)(34,45)(38,52)(39,49)(41,53)(43,47)(44,51)(48,50);(15,18)(19,21)(22,24,28,27)(23,29,32,42)(25,33,39,38)(26,34,41,36)(30,43,44,48)(31,40,45,53)(35,46,49,52)(37,47,51,50);(22,25,36,50,27,38,41,51,28,39,34,47,24,33,26,37)(23,30,40,46,29,43,45,49,32,44,53,52,42,48,31,35);(14,16,17,20)(15,19,18,21)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 26723);