Finite Group DatabasebetaFinite Group Databasebeta| order | 32766 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C127⋊C3)⋊D43 |
| Fitting, socle, derived, Hall | C43×C127 ⋊ C2×C3 |
| Hall | C43×C127⋊C3 ⋊ C2 |
| Hall | D5461 ⋊ C3 |
| Hall | C127⋊C3 ⋊ D43 |
| Sylow | C127 ⋊ C3×D43 |
| Sylow | C43 ⋊ D127⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C43×C127 | C2×C3 |
| LCS | G ▷ C43×C127 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 170 |
| transitive degree | μt(G) = 5461 |
A minimal faithful permutation representation — 2 generators on 170 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(45,49,50,46,47,51)(48,57,58,52,53,67)(54,70,71,60,61,84)(55,69,65,63,59,56)(62,89,90,73,74,107)(64,87,88,76,72,81)(66,68,86,78,79,82)(75,112,113,92,93,135)(77,110,111,95,91,104)(80,85,109,97,98,105)(83,124,122,101,102,100)(94,140,141,115,116,161)(96,138,139,117,114,132)(99,108,137,119,120,133)(103,148,149,125,126,128)(106,154,146,129,130,123)(118,163,164,143,142,159)(121,136,162,145,144,160)(127,166,167,150,151,157)(131,170,169,155,156,153)(134,168,165,152,158,147);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(44,45,48,54,62,75,94,116,142,151,156,158,160,161,145,119,97,78,65,50,49,55,64,77,96,118,144,93,114,126,130,133,135,159,165,146,122,88,58,69,57,68,83,103,127,152,120,74,91,102,105,107,132,157,169,149,111,71,86,87,70,85,106,131,155,129,98,61,72,82,84,104,128,153,167,139,90,109,124,110,89,108,134,150,125,101,79,53,59,67,81,100,123,147,164,113,137,154,148,138,112,136,143,117,95,76,63,47,51,56,66,80,99,121,141,162,168,170,166,163,140,115,92,73,60,52,46)Polycyclic presentation:
2,3,43,127|c^a=c^42,d^a=d^126,d^b=d^107972 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32766, 21);