Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13215 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C22×C4 |
| direct product | C17×#13216 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | #1961 |
| 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 | |
434 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)(20,31)(21,37,41,70)(22,43)(23,29)(24,47)(25,30)(26,34)(27,40,55,42)(28,44)(32,56)(33,63,65,79)(35,58)(36,60)(38,71,74,57)(39,75,76,54)(45,48)(46,59)(49,73,68,80)(50,66)(51,61)(52,67)(53,64,78,81)(62,77,72,69);(18,20,32,61,24,35,22,34,25,36,44,66,23,46,45,67)(19,26,48,58,29,51,28,31,30,52,43,59,47,50,56,60)(21,38,72,81,42,65,76,80,41,74,62,64,40,33,39,73)(27,53,69,71,37,68,75,63,55,78,77,57,70,49,54,79);(18,21)(19,27,30,55)(20,33,36,65)(22,39)(23,40)(24,42)(25,41)(26,49)(28,54,56,75)(29,37,47,70)(31,57)(32,62)(34,64,67,81)(35,38,46,74)(43,69,48,77)(44,72)(45,76)(50,78)(51,53)(52,68)(58,63)(59,79)(60,71)(61,80,66,73)578 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6204);