Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24987 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C17 · Schur multiplier (H2) = C24 |
| direct product | C17×#24989 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | #2420 |
| commutator, Frattini, cosocle | C22×C4 | C24×C17 |
| prime radical | C2×#41×C17 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C22×C4×C17 ◁ G | |
620 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 4 generators on 81 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,25,31)(20,32,37,50)(21,38,42,57)(22,43,46,55)(23,30,24,29)(26,36,51,35)(27,47,54,41)(28,45,56,44)(33,52,39,48)(34,62,65,49)(40,70,72,53)(58,66,67,61)(59,64,75,63)(60,79,69,78)(68,73,77,71)(74,80,76,81);(18,20,24,36,25,37,23,35)(19,26,30,50,31,51,29,32)(21,39,41,66,42,33,47,61)(22,34,45,64,46,65,44,63)(27,52,38,67,54,48,57,58)(28,49,55,75,56,62,43,59)(40,69,71,81,72,60,73,80)(53,76,68,79,70,74,77,78);(18,21)(19,27)(20,33)(22,40)(23,41)(24,47)(25,42)(26,48)(28,53)(29,38)(30,57)(31,54)(32,58)(34,60)(35,61)(36,66)(37,39)(43,68)(44,71)(45,73)(46,72)(49,74)(50,67)(51,52)(55,77)(56,70)(59,78)(62,76)(63,80)(64,81)(65,69)(75,79);(18,22,24,45,25,46,23,44)(19,28,30,55,31,56,29,43)(20,34,36,64,37,65,35,63)(21,40,47,73,42,72,41,71)(26,49,50,75,51,62,32,59)(27,53,57,77,54,70,38,68)(33,60,66,81,39,69,61,80)(48,74,67,79,52,76,58,78)680 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 17977);