Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C17 · Schur multiplier (H2) = C25 |
| direct product | C17×#33414 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | C22×D8 |
| commutator | C8 | C23×C4×C17 |
| Frattini, cosocle | C2×C8 | C24×C17 |
| socle | C2×C17 | C22×#107 |
| prime radical | C2×#41×C17 | D4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C2×#41×C17 ◁ C2×C17×#131 ◁ G | |
622 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)(18,19,23,29,25,30,24,45)(20,31,35,56,37,57,36,65)(21,27,40,52,38,50,41,69)(22,42,43,53,44,70,32,28)(26,46,48,64,49,58,55,34)(33,54,62,77,60,75,63,80)(39,67,68,74,66,81,59,51)(47,71,73,79,72,78,76,61);(18,20,32,58,24,36,44,64,25,37,43,46,23,35,22,34)(19,26,42,56,29,48,53,57,30,49,70,65,45,55,28,31)(21,33,59,78,41,63,66,79,38,60,68,71,40,62,39,61)(27,47,67,77,52,73,74,75,50,72,81,80,69,76,51,54);(18,21)(19,27)(20,33)(22,39)(23,40)(24,41)(25,38)(26,47)(28,51)(29,52)(30,50)(31,54)(32,59)(34,61)(35,62)(36,63)(37,60)(42,67)(43,68)(44,66)(45,69)(46,71)(48,73)(49,72)(53,74)(55,76)(56,77)(57,75)(58,78)(64,79)(65,80)(70,81);(21,38)(27,50)(33,60)(39,66)(40,41)(47,72)(51,74)(52,69)(54,75)(59,68)(61,79)(62,63)(67,81)(71,78)(73,76)(77,80)1088 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 26402);