Finite Group DatabasebetaFinite Group Databasebeta| order | 32766 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Hall | C43×C127 ⋊ C2×C3 |
| Hall | #30655814 ⋊ C2 |
| Hall | D5461 ⋊ C3 |
| Sylow | C127 ⋊ D43⋊C3 |
| 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:
(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,36)(12,26,22,20,16,13)(19,32,34,23,25,37)(21,38,39,30,29,33)(31,41,42,40,35,43)(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)(75,112,113,92,93,136)(77,110,111,95,91,104)(80,85,109,97,98,105)(83,125,123,101,102,100)(94,141,117,115,116,143)(96,139,140,118,114,133)(99,108,138,120,121,134)(103,151,152,126,127,129)(106,156,149,130,131,124)(119,164,148,145,142,162)(122,137,144,146,147,163)(128,168,167,153,154,160)(132,170,169,157,158,155)(135,165,166,159,161,150);(1,2,5,11,19,22,7,6,12,21,31,39,15,26,14,25,35,42,28,37,38,27,18,29,34,36,43,41,32,10,16,24,33,40,30,20,4,8,13,23,17,9,3)(44,45,48,54,62,75,94,117,144,165,170,168,164,141,145,118,95,76,63,47,51,56,66,80,99,122,148,113,138,156,151,139,112,137,153,126,101,79,53,59,67,81,100,124,150,167,140,90,109,125,110,89,108,135,157,130,98,61,72,82,84,104,129,155,169,152,111,71,86,87,70,85,106,132,159,121,74,91,102,105,107,133,160,166,149,123,88,58,69,57,68,83,103,128,147,93,114,127,131,134,136,162,146,120,97,78,65,50,49,55,64,77,96,119,116,142,154,158,161,163,143,115,92,73,60,52,46)Polycyclic presentation:
2,3,43,127|c^a=c^42,d^a=d^126,c^b=c^36,d^b=d^19916 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32766, 24);