Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C24 |
| direct product | C17×#13021 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #447 |
| commutator | C2×C8 | C22×C4×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| socle | C2×C17 | #1879 |
| prime radical | C17×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C22×C4×C17 ◁ C17×#405 ◁ G | |
462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 3 generators on 81 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,24,31,25,32,48,63,26,33,23,30,47,62,22,29)(20,34,37,56,38,61,36,66,39,64,46,27,51,54,49,55)(21,40,43,67,44,65,41,28,45,58,42,59,52,57,50,60)(35,69,71,76,72,53,77,78,73,79,74,81,75,80,70,68);(18,20,25,51,26,39,47,38)(19,27,32,66,33,56,62,55)(21,35,52,72,45,73,44,75)(22,46,24,36,48,37,23,49)(28,53,67,79,60,80,59,69)(29,61,31,34,63,54,30,64)(40,76,57,78,58,81,65,68)(41,71,43,77,50,74,42,70);(18,21)(19,28)(20,35)(22,43)(23,41)(24,50)(25,52)(26,45)(27,53)(29,58)(30,57)(31,65)(32,67)(33,60)(34,68)(36,74)(37,70)(38,75)(39,73)(40,63)(42,48)(44,47)(46,77)(49,71)(51,72)(54,76)(55,69)(56,80)(59,62)(61,81)(64,78)(66,79)782 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6009);