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×#33860 |
| 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 | |
636 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)(18,19,26,32)(20,33,38,68)(21,39,28,54)(22,43,46,42)(23,47,52,63)(24,31,25,30)(27,55,59,36)(29,50,64,51)(34,35,40,73)(37,75,53,61)(41,71,60,45)(44,78,56,57)(48,65,80,58)(49,62,81,79)(66,67,72,70)(69,74,77,76);(18,20,26,38)(19,27,32,59)(21,40,28,34)(22,44,46,56)(23,48,52,80)(24,37,25,53)(29,61,64,75)(30,58,31,65)(33,63,68,47)(35,71,73,45)(36,50,55,51)(39,76,54,74)(41,78,60,57)(42,67,43,70)(49,72,81,66)(62,69,79,77);(18,21,26,28)(19,22,32,46)(20,34,38,40)(23,41,52,60)(24,43,25,42)(27,56,59,44)(29,45,64,71)(30,54,31,39)(33,66,68,72)(35,61,73,75)(36,69,55,77)(37,67,53,70)(47,49,63,81)(48,57,80,78)(50,62,51,79)(58,76,65,74);(18,22)(19,28)(20,35)(21,32)(23,49)(24,39)(25,54)(26,46)(27,57)(29,62)(30,43)(31,42)(33,67)(34,61)(36,74)(37,72)(38,73)(40,75)(41,63)(44,48)(45,51)(47,60)(50,71)(52,81)(53,66)(55,76)(56,80)(58,69)(59,78)(64,79)(65,77)(68,70)731 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 26848);