Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C22×C4 |
| direct product | C13×#26203 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C13 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C13 |
| prime radical | C13×#2414 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C2×C4×D4×C13 ◁ G | |
844 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)(32,39)(36,44)(40,45)(41,43);(14,16,17,21)(15,18,19,20)(23,26)(27,28)(30,31)(32,40)(33,37)(34,38)(35,42)(36,41)(39,45)(43,44);(22,24,25,29)(23,27,26,28)(30,32,34,39)(31,36,38,44)(33,41,35,43)(37,45,42,40);(15,19)(18,20)(23,26)(27,28)(30,33,34,35)(31,37,38,42)(32,41,39,43)(36,45,44,40)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 19191);