Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33856 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C17 · Schur multiplier (H2) = C25 |
| direct product | C17×#33859 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | D4×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C17 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C2×C4×C17 ◁ C17×#443 ◁ G | |
646 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)(18,19,25,31)(20,32,37,67)(21,38,42,63)(22,43,48,61)(23,30,24,29)(26,51,55,35)(27,49,58,41)(28,46,62,47)(33,53,39,77)(34,52,74,56)(36,75,50,59)(40,78,72,57)(44,64,68,54)(45,60,79,81)(65,71,80,70)(66,73,76,69);(18,20,25,37)(19,26,31,55)(21,39,42,33)(22,44,48,68)(23,36,24,50)(27,56,58,52)(28,59,62,75)(29,54,30,64)(32,61,67,43)(34,72,74,40)(35,46,51,47)(38,76,63,66)(41,71,49,70)(45,73,79,69)(53,78,77,57)(60,80,81,65);(18,21)(19,27)(20,33)(22,40)(23,49)(24,41)(25,42)(26,52)(28,57)(29,63)(30,38)(31,58)(32,65)(34,68)(35,69)(36,71)(37,39)(43,60)(44,74)(45,47)(46,79)(48,72)(50,70)(51,73)(53,75)(54,76)(55,56)(59,77)(61,81)(62,78)(64,66)(67,80);(18,21)(19,27)(20,34)(22,45)(23,41)(24,49)(25,42)(26,53)(28,60)(29,38)(30,63)(31,58)(32,66)(33,68)(35,70)(36,73)(37,74)(39,44)(40,47)(43,57)(46,72)(48,79)(50,69)(51,71)(52,75)(54,80)(55,77)(56,59)(61,78)(62,81)(64,65)(67,76)731 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 26847);