Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C17 · Schur multiplier (H2) = C25 |
| direct product | C17×#33617 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #466 |
| commutator, Frattini, cosocle | C2×C8 | C24×C17 |
| socle | C2×C17 | C2×#466 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C22×C4×C17 ◁ C2×C17×#131 ◁ G | |
642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 4 generators on 81 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,32)(21,39)(22,28)(23,45)(24,30)(25,31)(26,36)(27,42)(29,33)(34,62)(35,57)(37,58)(38,59)(40,52)(41,72)(43,55)(44,54)(46,53)(47,74)(48,67)(49,60)(50,68)(51,73)(56,65)(61,70)(63,78)(64,69)(66,79)(71,77)(75,80)(76,81);(18,20,33,60,24,37,46,68,25,38,44,67,22,35,23,36)(19,26,45,57,28,48,54,59,31,50,53,58,30,49,29,32)(21,34,61,74,42,65,73,81,43,66,71,80,40,63,41,64)(27,47,70,62,39,69,72,78,52,75,77,79,55,76,51,56);(18,21)(19,27)(20,34)(22,40)(23,41)(24,42)(25,43)(26,47)(28,39)(29,51)(30,55)(31,52)(32,56)(33,61)(35,63)(36,64)(37,65)(38,66)(44,71)(45,70)(46,73)(48,69)(49,76)(50,75)(53,77)(54,72)(57,62)(58,79)(59,78)(60,74)(67,80)(68,81);(18,22,25,24)(19,28,31,30)(20,35,38,37)(21,40,43,42)(23,44,46,33)(26,48,50,49)(27,39,52,55)(29,45,54,53)(32,57,59,58)(34,63,66,65)(36,67,68,60)(41,71,73,61)(47,69,75,76)(51,70,72,77)(56,62,78,79)(64,80,81,74)884 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 26605);