Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #62636 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C17 · Schur multiplier (H2) = C29 |
| direct product | C17×#62637 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | C23×D8 |
| commutator, Frattini, cosocle | C8 | C25×C17 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C2×#41×C17 ◁ C2×C17×#131 ◁ G | |
2026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 5 generators on 81 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,30)(21,26)(22,38)(23,43)(24,27)(25,32)(28,41)(29,40)(31,55)(33,62)(34,56)(35,48)(36,65)(37,68)(39,53)(42,54)(44,52)(45,58)(46,61)(47,70)(49,67)(50,64)(51,60)(57,77)(59,80)(63,76)(66,75)(69,74)(71,78)(72,81)(73,79);(18,20,24,34)(19,25,27,46)(21,31,35,57)(22,39,42,70)(23,44,40,33)(26,45,48,71)(28,52,51,62)(29,53,43,47)(30,54,56,38)(32,60,61,41)(36,66,63,81)(37,69,64,59)(49,74,73,80)(50,75,68,72)(55,76,77,65)(58,79,78,67);(18,21)(19,26)(20,31)(22,36)(23,37)(24,35)(25,45)(27,48)(28,49)(29,50)(30,55)(32,58)(33,59)(34,57)(38,65)(39,66)(40,64)(41,67)(42,63)(43,68)(44,69)(46,71)(47,72)(51,73)(52,74)(53,75)(54,76)(56,77)(60,79)(61,78)(62,80)(70,81);(21,35)(26,48)(31,57)(36,63)(37,64)(45,71)(49,73)(50,68)(55,77)(58,78)(59,69)(65,76)(66,81)(67,79)(72,75)(74,80);(19,27)(25,46)(26,48)(28,51)(29,43)(30,56)(38,54)(45,71)(47,53)(49,73)(50,68)(52,62)(55,77)(65,76)(72,75)(74,80)986 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 55625);