Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C17 · Schur multiplier (H2) = C29 |
| direct product | C17×#62635 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | C22×D8 |
| commutator, Frattini, cosocle | C8 | C25×C17 |
| socle | C2×C17 | C23×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C2×#41×C17 ◁ C2×C17×#131 ◁ G | |
2038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 5 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)(21,27)(22,28)(23,41)(24,45)(25,30)(26,34)(29,43)(32,56)(33,57)(35,58)(36,51)(37,52)(38,65)(39,68)(40,55)(42,48)(44,70)(46,54)(47,61)(49,71)(50,62)(53,67)(59,77)(60,78)(63,80)(64,75)(66,73)(69,76)(72,79)(74,81);(18,20)(19,26)(21,32)(22,33)(23,42)(24,46)(25,35)(27,47)(28,48)(29,54)(30,50)(31,41)(34,43)(36,59)(37,60)(38,66)(39,69)(40,62)(44,71)(45,49)(51,72)(52,73)(53,76)(55,57)(56,65)(58,70)(61,67)(63,81)(64,79)(68,74)(75,78)(77,80);(18,21)(19,27)(20,32)(22,37)(23,38)(24,39)(25,36)(26,47)(28,52)(29,53)(30,51)(31,56)(33,60)(34,61)(35,59)(40,64)(41,65)(42,66)(43,67)(44,63)(45,68)(46,69)(48,73)(49,74)(50,72)(54,76)(55,75)(57,78)(58,77)(62,79)(70,80)(71,81);(21,36)(27,51)(32,59)(37,39)(38,63)(47,72)(52,68)(53,75)(56,77)(60,69)(61,79)(64,67)(65,80)(66,81)(73,74)(76,78);(18,22,25,24)(19,28,30,45)(20,33,35,46)(21,37,36,39)(23,40,44,43)(26,48,50,49)(27,52,51,68)(29,41,55,70)(31,57,58,54)(32,60,59,69)(34,42,62,71)(38,64,63,67)(47,73,72,74)(53,65,75,80)(56,78,77,76)(61,66,79,81)1088 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 55623);