Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#32667 |
| 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 ◁ C2×C4×Q8×C13 ◁ G | |
666 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)(14,15,18,20)(16,17,21,19)(22,23)(24,28)(25,27)(26,29)(30,31,34,36)(32,39,41,37)(33,42,35,38)(40,44,43,45);(14,16,18,21)(15,19,20,17)(22,24)(23,26)(25,29)(27,28)(31,36)(32,40)(33,35)(37,44)(39,45)(41,43);(23,27)(26,28)(30,32,34,41)(31,37,36,39)(33,40,35,43)(38,44,42,45);(14,17,18,19)(15,16,20,21)(24,29)(26,28)(30,33,34,35)(31,38,36,42)(32,40,41,43)(37,44,39,45)520 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 25655);