Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11669 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C25 |
| direct product | C17×#12746 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #418 |
| 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)(18,19,23,30,25,29,24,47)(20,31,35,60,34,61,48,56)(21,36,38,55,39,27,46,54)(22,40,43,58,45,72,44,28)(26,41,50,67,52,74,51,33)(32,59,64,81,63,77,65,80)(37,69,71,57,42,75,76,53)(49,70,78,62,68,73,79,66);(18,20)(19,26)(21,32)(22,41)(23,35)(24,48)(25,34)(27,49)(28,56)(29,52)(30,50)(31,40)(33,44)(36,68)(37,70)(38,64)(39,63)(42,73)(43,67)(45,74)(46,65)(47,51)(53,80)(54,78)(55,79)(57,81)(58,60)(59,69)(61,72)(62,71)(66,76)(75,77);(18,21)(19,27)(20,32)(22,42)(23,46)(24,38)(25,39)(26,49)(28,57)(29,36)(30,55)(31,59)(33,66)(34,63)(35,65)(37,45)(40,69)(41,73)(43,71)(44,76)(47,54)(48,64)(50,79)(51,78)(52,68)(53,58)(56,81)(60,80)(61,77)(62,67)(70,74)(72,75)782 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5734);