Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#29090 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C13 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C13 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C4×C13×C4⋊C4 ◁ G | |
642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 3328 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,17,18)(16,20,21,19)(22,23,25,27)(24,28,29,26)(30,31)(32,36)(33,41)(34,37)(35,38)(39,43)(40,44)(42,45);(15,18)(19,20)(22,24,25,29)(23,26,27,28)(30,32,39,42,34,33,35,40)(31,36,43,45,37,41,38,44);(23,27)(26,28)(30,33,34,32)(31,36,37,41)(35,40,39,42)(38,45,43,44);(14,16,17,21)(15,19,18,20)(23,27)(26,28)(31,37)(36,41)(38,43)(44,45)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 22078);