Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C24 |
| direct product | C13×#33685 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C13 |
| prime radical | C13×#2186 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C13×#117 ◁ C13×#405 ◁ G | |
546 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)(22,23,25,30)(24,33,34,28)(26,29,37,44)(27,40,38,32)(31,46,45,39)(35,47,50,41)(36,43,51,49)(42,52,48,53);(15,18)(19,21)(22,24,25,34)(23,28,30,33)(26,35,37,50)(27,36,38,51)(29,41,44,47)(31,42,45,48)(32,43,40,49)(39,52,46,53);(23,29)(26,39)(27,38)(28,41)(30,44)(31,32)(33,47)(35,52)(36,51)(37,46)(40,45)(42,43)(48,49)(50,53);(14,16,17,20)(15,19,18,21)(24,34)(28,33)(35,50)(36,51)(41,47)(42,48)(43,49)(52,53)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 26673);