Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13312 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C17 · Schur multiplier (H2) = C23 |
| direct product | C17×#13406 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #469 |
| 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)(20,31)(21,37)(22,29)(23,45)(24,30)(25,33)(26,40)(27,44)(28,43)(32,63)(34,50)(35,52)(36,51)(38,71)(39,72)(41,58)(42,70)(46,57)(47,56)(48,59)(49,68)(53,79)(54,75)(55,76)(60,74)(61,65)(62,69)(64,81)(66,80)(67,78)(73,77);(18,20)(19,25)(21,38)(22,35)(23,46)(24,36)(26,53)(27,51)(28,57)(29,31)(30,52)(32,41)(33,44)(34,48)(37,63)(39,66)(40,73)(42,64)(43,67)(45,62)(47,65)(49,54)(50,60)(55,61)(56,77)(58,81)(59,78)(68,76)(69,74)(70,80)(71,72)(75,79);(18,21,24,42)(19,26,30,55)(20,32)(22,41,44,39)(23,47,48,75)(25,49)(27,56,29,54)(28,58,60,72)(31,61)(33,38)(34,68)(35,64)(36,66)(37,59,70,45)(40,74,76,43)(46,73)(50,63)(51,53)(52,77)(57,80)(62,81)(65,69)(67,79)(71,78)680 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6394);