Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#26211 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C13 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C13 |
| prime radical | C22×D4×C13 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C2×C4×D4×C13 ◁ G | |
780 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1664 |
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)(16,20)(17,19)(18,21)(22,23)(24,28)(25,26)(27,29)(30,31,34,35)(32,36,39,43)(33,40,42,37)(38,41,45,44);(14,16,17,21)(15,18,19,20)(23,26)(27,28)(30,32)(31,36)(33,41)(34,39)(35,43)(37,38)(40,45)(42,44);(22,24,25,29)(23,27,26,28)(30,33,34,42)(31,37,35,40)(32,38,39,45)(36,44,43,41);(15,19)(18,20)(23,26)(27,28)(30,31,34,35)(32,36,39,43)(33,37,42,40)(38,44,45,41)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 19199);