Finite Group DatabasebetaFinite Group Databasebeta| order | 32526 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#24925704 |
| Fitting, socle, derived | C3×C13×C139 ⋊ C2×C3 |
| Hall | C3×#19129429 ⋊ C2 |
| Hall | C13×C139 ⋊ C3×D3 |
| Sylow | C139 ⋊ D3×C13⋊C3 |
| Sylow | C13 ⋊ C3⋊(D139⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C13×C139 | C2×C3 |
| LCS | G ▷ C3×C13×C139 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 155 |
| transitive degree | μt(G) = 5421 |
A minimal faithful permutation representation — 2 generators on 155 points:
(2,3)(5,6,8)(7,10,13)(9,12,11)(14,16,15)(18,22,23,19,20,24)(21,30,31,25,26,40)(27,43,44,33,34,57)(28,42,38,36,32,29)(35,62,63,46,47,80)(37,60,61,49,45,54)(39,41,59,51,52,55)(48,85,86,65,66,109)(50,83,84,68,64,77)(53,58,82,70,71,78)(56,98,96,74,75,73)(67,114,115,88,89,138)(69,112,113,91,87,106)(72,81,111,93,94,107)(76,125,126,99,100,102)(79,129,123,103,104,97)(90,130,128,105,117,148)(92,141,142,118,116,135)(95,110,140,120,121,136)(101,143,147,119,127,131)(108,150,145,132,133,124)(122,139,149,134,144,154)(137,155,153,151,152,146);(1,2,3)(4,5,7,6,9,13,10,14,16,8,11,15,12)(17,18,21,27,35,48,67,90,103,71,34,45,55,57,77,102,128,149,155,144,89,116,127,130,133,136,138,147,113,63,82,98,83,62,81,108,134,120,93,70,51,38,23,22,28,37,50,69,92,119,99,74,52,26,32,40,54,73,97,124,146,151,121,66,87,100,104,107,109,135,142,86,111,129,125,112,85,110,137,153,145,123,96,61,31,42,30,41,56,76,101,118,91,68,49,36,20,24,29,39,53,72,95,122,132,94,47,64,75,78,80,106,131,115,140,150,117,143,141,114,139,152,154,148,126,84,44,59,60,43,58,79,105,88,65,46,33,25,19)Polycyclic presentation:
2,3^2,13,139|c^a=c^2,e^a=e^138,d^b=d^3,e^b=e^96918 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32526, 52);