Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C24 |
| direct product | C13×#13020 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #447 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| prime radical | C23×C13 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ C13×#405 ◁ G | |
460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 3 generators on 81 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(15,17)(18,19,24,31,48,64,49,30,26,33,22,29,25,32,23,50)(20,34,38,57,52,55,37,27,40,56,36,66,39,63,51,58)(21,41,44,62,53,59,47,61,46,65,42,28,45,60,43,67)(35,69,72,68,76,81,71,77,74,80,70,78,73,54,75,79);(14,15,16,17)(18,20)(19,27)(21,35)(22,37)(23,38)(24,51)(25,52)(26,40)(28,54)(29,63)(30,34)(31,55)(32,66)(33,58)(36,49)(39,48)(41,77)(42,71)(43,72)(44,75)(45,76)(46,74)(47,70)(50,56)(53,73)(57,64)(59,68)(60,78)(61,69)(62,81)(65,79)(67,80);(18,21,26,46)(19,28,33,62)(20,35,40,74)(22,47,24,43)(23,44,49,42)(25,53,48,45)(27,54,58,81)(29,41,31,65)(30,59,50,60)(32,67,64,61)(34,68,56,78)(36,71,38,75)(37,70,51,72)(39,76,52,73)(55,79,63,77)(57,69,66,80)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6008);