Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13030 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C24 |
| direct product | C17×#13031 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | #1879 |
| commutator | C2×C8 | C22×C4×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| prime radical | C2×C17×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C22×C4×C17 ◁ C17×#405 ◁ G | |
456 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)(18,19,23,29,48,51,46,28,25,31,45,62,24,30,22,44)(20,32,35,26,49,53,34,63,37,66,73,55,36,60,47,54)(21,38,41,59,50,61,39,58,43,56,76,27,42,57,40,64)(33,52,69,65,74,81,68,77,71,80,72,75,70,79,67,78);(18,20,24,49,25,37,48,36)(19,26,51,54,31,55,30,63)(21,33,50,70,43,71,42,74)(22,35,45,34,46,73,23,47)(27,52,58,81,59,80,64,79)(28,60,62,66,44,53,29,32)(38,75,57,78,56,65,61,77)(39,72,76,68,40,69,41,67);(18,21)(19,27)(20,33)(22,41)(23,40)(24,50)(25,43)(26,52)(28,61)(29,56)(30,64)(31,59)(32,65)(34,72)(35,67)(36,74)(37,71)(38,62)(39,45)(42,48)(44,57)(46,76)(47,69)(49,70)(51,58)(53,78)(54,81)(55,80)(60,77)(63,79)(66,75)(68,73)680 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6019);