Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C2×C42 |
| direct product | C13×#32673 |
| 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 ◁ C2×C4×Q8×C13 ◁ G | |
738 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)(14,15,18,20)(16,19,17,21)(23,26)(24,28)(30,31,34,37)(32,35,40,44)(33,36,41,45)(38,42,39,43);(14,15)(16,19)(17,21)(18,20)(22,23,25,26)(24,27,28,29)(30,32)(31,35)(33,38)(34,40)(36,42)(37,44)(39,41)(43,45);(14,16,18,17)(15,19,20,21)(22,24,25,28)(23,27,26,29)(32,38)(33,41)(35,42)(36,45)(39,40)(43,44);(14,17,18,16)(15,19,20,21)(23,26)(27,29)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 25661);