Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#27018 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C13 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C13 |
| prime radical | C22×D4×C13 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C4×C13×C4⋊C4 ◁ G | |
706 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)(14,15,18,21)(16,20,17,19)(22,23,25,27)(24,28,29,26)(30,31,34,39)(32,40,42,36)(33,37,35,38)(41,44,43,45);(14,16,18,17)(15,19,21,20)(22,24,25,29)(23,26,27,28)(30,32)(31,36)(33,41)(34,42)(35,43)(37,45)(38,44)(39,40);(23,27)(26,28)(30,33)(31,37)(34,35)(38,39)(41,43)(44,45);(14,17,18,16)(15,20,21,19)(23,27)(26,28)(30,33,34,35)(31,38,39,37)(32,41,42,43)(36,44,40,45)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 20006);