Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#13456 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C13 |
| prime radical | C23×C13 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ C13×#405 ◁ G | |
318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,18,20)(16,21,17,19)(22,23,25,30)(24,34,36,28)(26,29,39,45)(27,32,40,33)(31,47,46,41)(35,43,38,48)(37,52,51,44)(42,50,49,53);(14,16,18,17)(15,19,20,21)(22,24,25,36)(23,28,30,34)(26,37,39,51)(27,35,40,38)(29,44,45,52)(31,42,46,49)(32,48,33,43)(41,50,47,53);(14,17,18,16)(15,19,20,21)(23,29)(24,35)(26,41)(27,40)(28,42)(30,45)(31,33)(32,46)(34,49)(36,38)(37,51)(39,47)(43,52)(44,48)520 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6444);