Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13068 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C24 |
| direct product | C17×#13069 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | #1893 |
| commutator | C2×C8 | C22×C4×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| prime radical | C2×#41×C17 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C22×C4×C17 ◁ C2×C17×#131 ◁ G | |
450 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)(21,37,40,55)(22,41)(23,29)(24,47)(25,30)(26,34)(27,39,54,46)(28,43)(32,57)(33,64,67,80)(35,59)(36,61)(38,53,42,56)(44,48)(45,60)(49,76,79,81)(50,70)(51,62)(52,71)(58,66,77,68)(63,73,74,75)(65,72,69,78);(18,20,32,62,24,35,22,34,25,36,43,70,23,45,44,71)(19,26,48,59,29,51,28,31,30,52,41,60,47,50,57,61)(21,33,63,76,39,68,42,69,40,67,74,81,46,66,38,65)(27,49,75,80,55,78,56,58,54,79,73,64,37,72,53,77);(18,21)(19,27)(20,33)(22,42)(23,46)(24,39)(25,40)(26,49)(28,56)(29,55)(30,54)(31,58)(32,63)(34,69)(35,68)(36,67)(37,47)(38,44)(41,73)(43,74)(45,66)(48,75)(50,72)(51,78)(52,79)(53,57)(59,80)(60,64)(61,77)(62,76)(65,71)(70,81)782 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6057);