Finite Group DatabasebetaFinite Group Databasebeta| order | 26586 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#22579723 |
| Fitting, socle, derived | C3×C7×C211 ⋊ C2×C3 |
| Hall | C3×#17984522 ⋊ C2 |
| Hall | C7×C211 ⋊ C3×D3 |
| Sylow | C211 ⋊ D3×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C3⋊(D211⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C7×C211 | C2×C3 |
| LCS | G ▷ C3×C7×C211 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 221 |
| transitive degree | μt(G) = 4431 |
A minimal faithful permutation representation — 2 generators on 221 points:
(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,41)(12,26,22,20,16,13)(19,46,47,30,31,64)(21,44,45,33,29,38)(23,25,43,35,36,39)(32,69,70,49,50,93)(34,67,68,52,48,61)(37,42,66,54,55,62)(40,82,80,58,59,57)(51,98,99,72,73,128)(53,96,97,75,71,90)(56,65,95,77,78,91)(60,111,112,83,84,86)(63,117,109,87,88,81)(74,133,134,101,102,164)(76,131,132,104,100,125)(79,94,130,106,107,126)(85,141,142,113,114,121)(89,148,146,118,119,116)(92,153,139,122,123,110)(103,138,162,108,129,165)(105,166,167,136,135,161)(115,170,171,143,144,157)(120,177,178,149,150,152)(124,183,175,154,155,147)(127,188,169,158,159,140)(137,193,194,168,163,195)(145,196,197,172,173,192)(151,198,199,179,180,187)(156,202,201,184,185,182)(160,207,174,189,190,176)(181,208,209,200,191,210)(186,211,205,203,204,206)(213,214,215)(216,218,217)(220,221);(1,2,5,11,19,32,51,74,103,106,77,54,35,22,7,6,12,21,34,53,76,105,137,169,139,109,80,45,15,26,14,25,40,60,85,115,145,174,175,146,112,68,28,43,44,27,42,63,89,120,151,181,201,178,142,97,47,66,82,67,46,65,92,124,156,186,205,199,171,132,70,95,117,111,96,69,94,127,160,191,204,209,197,167,99,130,153,148,141,131,98,129,163,173,180,185,190,194,134,165,188,183,177,170,166,133,102,135,144,150,155,159,162,164,195,207,202,198,196,193,138,73,100,114,119,123,126,128,161,192,210,211,208,189,158,107,50,71,84,88,91,93,125,157,187,206,203,184,154,122,78,31,48,59,62,64,90,121,152,182,200,179,149,118,87,55,18,29,39,41,61,86,116,147,176,172,143,113,83,58,36,10,16,24,38,57,81,110,140,168,136,104,75,52,33,20,4,8,13,23,37,56,79,108,101,72,49,30,17,9,3)(212,213,214,216,215,217,218)(219,220,221)Polycyclic presentation:
2,3^2,7,211|c^a=c^2,e^a=e^210,d^b=d^2,e^b=e^196750 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26586, 68);