Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C24 |
| direct product | C17×#13020 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C17 | #447 |
| commutator | C2×C8 | C22×C4×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| prime radical | C23×C17 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C17 ◁ C22×C4×C17 ◁ C17×#405 ◁ G | |
460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 2176 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(19,21)(22,23,28,35,52,68,53,34,30,37,26,33,29,36,27,54)(24,38,42,61,56,59,41,31,44,60,40,70,43,67,55,62)(25,45,48,66,57,63,51,65,50,69,46,32,49,64,47,71)(39,73,76,72,80,85,75,81,78,84,74,82,77,58,79,83);(18,19,20,21)(22,24)(23,31)(25,39)(26,41)(27,42)(28,55)(29,56)(30,44)(32,58)(33,67)(34,38)(35,59)(36,70)(37,62)(40,53)(43,52)(45,81)(46,75)(47,76)(48,79)(49,80)(50,78)(51,74)(54,60)(57,77)(61,68)(63,72)(64,82)(65,73)(66,85)(69,83)(71,84);(22,25,30,50)(23,32,37,66)(24,39,44,78)(26,51,28,47)(27,48,53,46)(29,57,52,49)(31,58,62,85)(33,45,35,69)(34,63,54,64)(36,71,68,65)(38,72,60,82)(40,75,42,79)(41,74,55,76)(43,80,56,77)(59,83,67,81)(61,73,70,84)782 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6008);