Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C25 |
| direct product | C17×#12747 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #447 |
| commutator | C2×C8 | C22×C4×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| socle | C2×C17 | #1786 |
| prime radical | C17×#3584 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C22×C4×C17 ◁ C2×C17×#131 ◁ G | |
886 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)(21,36,39,54)(22,40)(23,29)(24,46)(25,30)(26,33)(27,38,53,45)(28,43)(32,60,63,80)(34,58)(35,59)(37,52,70,78)(41,49)(42,56)(44,72)(47,55)(48,75,76,68)(50,73)(51,65)(57,62,79,64)(61,66,81,77)(67,74,71,69);(18,20)(19,26)(21,32)(22,41)(23,34)(24,47)(25,35)(27,48)(28,55)(29,49)(30,51)(31,40)(33,43)(36,66)(37,68)(38,62)(39,63)(42,65)(44,73)(45,64)(46,50)(52,60)(53,76)(54,77)(56,58)(57,71)(59,72)(61,69)(67,79)(70,75)(74,81)(78,80);(18,21)(19,27)(20,32)(22,37)(23,45)(24,38)(25,39)(26,48)(28,52)(29,54)(30,53)(31,57)(33,61)(34,64)(35,63)(36,46)(40,71)(41,68)(42,74)(43,69)(44,70)(47,62)(49,77)(50,66)(51,76)(55,60)(56,78)(58,80)(59,79)(65,81)(67,72)(73,75)782 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5735);