Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C17 · Schur multiplier (H2) = C23 |
| direct product | C17×#13401 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #482 |
| commutator, Frattini, cosocle | C22×C8 | C23×C17 |
| socle | C2×C17 | #2034 |
| prime radical | C17×#3067 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C22×C4×C17 ◁ C2×C17×#131 ◁ G | |
390 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)(20,30)(21,36)(22,41)(23,45)(24,29)(25,32)(26,39)(27,44)(28,43)(31,59)(33,49)(34,50)(35,51)(37,70)(38,72)(40,69)(42,55)(46,54)(47,68)(48,64)(52,75)(53,74)(56,73)(57,61)(58,65)(60,81)(62,67)(63,76)(66,80)(71,78)(77,79);(18,20)(19,25)(21,37)(22,34)(23,33)(24,35)(26,48)(27,51)(28,49)(29,50)(30,41)(31,40)(32,44)(36,67)(38,60)(39,71)(42,62)(43,65)(45,76)(46,77)(47,66)(52,57)(53,79)(54,78)(55,59)(56,80)(58,68)(61,74)(63,73)(64,75)(69,81)(70,72);(18,21,24,40)(19,26,29,53)(20,31)(22,42,44,38)(23,46,47,75)(25,48)(27,54,41,52)(28,55,56,72)(30,57)(32,62)(33,64)(34,60)(35,37)(36,68,69,45)(39,73,74,43)(49,59)(50,79)(51,78)(58,81)(61,65)(63,71)(66,77)(67,76)(70,80)680 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6389);