Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13229 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C22×C4 |
| direct product | C17×#13233 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | #1964 |
| commutator | C22×C4 | C22×C4×C17 |
| Frattini, cosocle | C2×#41 | C23×C17 |
| prime radical | C2×#41×C17 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C2×C4×C17 ◁ C2×C17×#131 ◁ G | |
432 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)(18,19,24,29,25,30,23,45)(20,31,35,58,36,59,46,57)(21,37,40,54,41,68,39,27)(22,42,32,55,44,28,47,49)(26,48,51,43,52,65,75,34)(33,61,63,56,64,78,72,79)(38,69,60,74,71,67,73,53)(50,62,66,76,77,70,81,80);(18,20,32,48)(19,26,49,58)(21,33,60,76)(22,43,24,46)(23,35,44,34)(25,36,47,65)(27,50,69,78)(28,31,29,51)(30,52,55,57)(37,66,53,56)(38,70,40,72)(39,63,71,62)(41,64,73,80)(42,59,45,75)(54,77,67,61)(68,81,74,79);(18,21,24,40,25,41,23,39)(19,27,45,68,30,54,29,37)(20,33,46,72,36,64,35,63)(22,38,47,73,44,71,32,60)(26,50,75,81,52,77,51,66)(28,53,49,69,42,74,55,67)(31,56,58,78,59,79,57,61)(34,62,48,76,43,70,65,80)578 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6221);