Finite Group DatabasebetaFinite Group Databasebeta| order | 26586 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#22579732 |
| Fitting, socle, derived | C3×C7×C211 ⋊ C2×C3 |
| Hall | C3×#17984521 ⋊ C2 |
| Hall | C7×C211 ⋊ C3×D3 |
| Sylow | C211 ⋊ C3⋊AGL(1,𝔽7) |
| 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)(74,133,134,101,102,165)(76,131,132,104,100,125)(79,94,130,106,107,126)(85,142,143,113,114,121)(89,149,147,118,119,116)(92,154,140,122,123,110)(103,135,162,105,136,168)(108,129,167,137,138,163)(115,172,173,144,145,158)(120,176,177,150,151,153)(124,182,174,155,156,148)(127,187,170,159,160,141)(139,166,191,146,169,195)(152,198,199,178,179,186)(157,202,200,183,184,181)(161,204,196,188,189,175)(164,208,180,192,193,171)(185,209,210,197,194,203)(190,211,207,205,206,201)(213,216,217,214,215,218)(220,221);(1,2,5,11,19,32,51,74,103,104,75,52,33,20,4,8,13,23,37,56,79,108,139,144,113,83,58,36,10,16,24,38,57,81,110,141,171,178,150,118,87,55,18,29,39,41,61,86,116,148,175,197,183,155,122,78,31,48,59,62,64,90,121,153,181,201,205,188,159,107,50,71,84,88,91,93,125,158,186,203,211,209,192,138,73,100,114,119,123,126,128,162,191,208,204,202,198,169,102,135,145,151,156,160,163,165,134,167,187,182,176,172,136,133,166,179,184,189,193,195,168,99,130,154,149,142,131,98,129,164,194,206,210,199,173,132,70,95,117,111,96,69,94,127,161,190,207,200,177,143,97,47,66,82,67,46,65,92,124,157,185,196,174,147,112,68,28,43,44,27,42,63,89,120,152,180,170,140,109,80,45,15,26,14,25,40,60,85,115,146,137,106,77,54,35,22,7,6,12,21,34,53,76,105,101,72,49,30,17,9,3)(212,213,215,218,217,216,214)(219,220,221)Polycyclic presentation:
2,3^2,7,211|c^a=c^2,d^a=d^6,e^a=e^210,d^b=d^2,e^b=e^14747 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26586, 61);