Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8491 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C24 |
| direct product | C17×#8501 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C17 | #379 |
| commutator | C42 | C22×C4×C17 |
| Frattini, cosocle | C2×C42 | C23×C17 |
| prime radical | C17×#121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C17 ◁ C2×C42×C17 ◁ G | |
616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 3 generators on 81 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,22,29,25,20,31,35)(21,37,40,70,42,62,72,69)(23,28,44,56,48,34,59,66)(24,30,45,27,49,36,60,33)(26,50,52,79,32,61,51,46)(38,71,41,55,43,76,75,65)(39,53,73,58,74,63,78,68)(47,57,77,54,80,67,81,64);(18,20)(19,25)(21,32)(22,27)(23,28)(24,36)(26,42)(29,45)(30,49)(31,33)(34,48)(35,60)(37,39)(38,51)(40,58)(41,63)(43,52)(44,64)(46,55)(47,57)(50,69)(53,75)(54,59)(56,81)(61,70)(62,74)(65,79)(66,77)(67,80)(68,72)(71,73)(76,78);(18,21,24,41)(19,26,30,53)(20,32,36,63)(22,43,45,40)(23,46,47,39)(25,42,49,75)(27,52,29,58)(28,55,57,37)(31,38,60,72)(33,51,35,68)(34,65,67,62)(44,73,77,50)(48,79,80,74)(54,76,56,70)(59,78,81,61)(64,71,66,69)578 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 1489);