Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C24 |
| direct product | C17×#13071 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #466 |
| commutator | C2×C8 | C22×C4×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| socle | C2×C17 | #1893 |
| prime radical | C17×#3067 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C22×C4×C17 ◁ C2×C17×#131 ◁ G | |
456 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)(20,31,36,56)(21,37,40,55)(22,41)(23,29)(24,48)(25,30)(26,34,51,42)(27,39,54,47)(28,32)(33,64)(35,60,46,61)(38,53,43,57)(44,58)(45,74)(49,75)(50,70,52,62)(59,67)(63,79,73,72)(65,77)(66,76)(68,80)(69,71)(78,81);(18,20,32,62,23,46,22,42,25,36,44,70,24,35,45,34)(19,26,41,60,48,50,28,56,30,51,74,61,29,52,58,31)(21,33,63,75,47,66,43,65,40,68,73,81,39,67,38,69)(27,49,72,80,37,71,57,76,54,78,79,64,55,77,53,59);(18,21)(19,27)(20,33)(22,43)(23,47)(24,39)(25,40)(26,49)(28,57)(29,55)(30,54)(31,59)(32,63)(34,69)(35,67)(36,68)(37,48)(38,45)(41,72)(42,65)(44,73)(46,66)(50,71)(51,78)(52,77)(53,58)(56,76)(60,80)(61,64)(62,75)(70,81)(74,79)884 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6059);