Finite Group DatabasebetaFinite Group Databasebeta| order | 26586 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊(C3⋊(D211⋊C7)) |
| unambiguous semidirect product | (C211⋊C7)⋊(C3⋊D3) |
| unambiguous semidirect product | (C3⋊D633)⋊C7 |
| Fitting, socle, derived, Hall | C32×C211 ⋊ C2×C7 |
| Hall | C32×C211⋊C7 ⋊ C2 |
| Hall | C3⋊D633 ⋊ C7 |
| Hall | C211⋊C7 ⋊ C3⋊D3 |
| Sylow | C211 ⋊ C7×C3⋊D3 |
| Sylow | C32 ⋊ D211⋊C7 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C32×C211 | C2×C7 |
| LCS | G ▷ C32×C211 | |
| UCS | C1 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 217 |
| transitive degree | μt(G) = 1899 |
A minimal faithful permutation representation — 3 generators on 217 points:
(2,6,7,25,30,106,123,3,4,12,14,54,64,172)(5,19,20,86,80,135,192,9,10,41,43,74,142,189)(8,35,36,88,89,182,186,16,17,75,42,153,154,188)(11,48,49,158,162,44,40,22,23,94,96,207,85,84)(13,59,60,24,101,102,77,27,28,114,51,52,164,138)(15,69,70,45,46,104,157,32,33,130,21,53,92,168)(18,81,82,194,196,87,202,38,39,143,144,204,150,151)(26,110,111,190,58,161,179,56,57,176,136,95,113,208)(29,120,121,166,205,90,137,62,63,127,184,210,76,156)(31,125,126,122,65,103,163,66,67,187,185,124,97,167)(34,129,134,68,107,108,201,72,73,112,128,55,173,149)(37,139,140,174,175,115,206,78,79,171,109,141,180,181)(47,148,159,61,131,98,99,91,93,200,117,118,50,100)(71,116,83,199,155,169,170,132,133,146,147,203,105,178)(119,209,198,177,197,152,191,165,183,145,211,193,195,160)(213,214)(216,217);(1,2,5,11,24,53,105,171,63,122,96,104,106,161,144,197,79,141,157,206,133,114,162,108,135,176,211,57,112,178,167,198,143,195,150,20,44,84,38,16,6,13,29,39,83,148,154,180,28,61,119,52,103,94,160,190,36,77,123,186,70,131,124,54,73,128,66,25,55,109,102,166,199,196,203,153,201,43,91,45,19,42,90,17,37,80,72,32,12,26,58,23,50,100,151,189,156,205,179,200,187,209,121,125,33,71,132,69,67,127,183,126,159,208,210,137,192,202,99,98,48,95,56,7,15,34,74,78,35,76,88,10,21,47,86,149,89,155,204,147,184,164,174,107,14,31,68,129,30,65,118,130,188,172,138,75,136,191,49,97,101,165,117,59,115,182,93,146,81,62,27,4,8,18,40,85,41,87,152,82,145,163,170,134,110,177,111,142,173,207,60,116,181,168,175,139,193,194,113,64,92,158,185,120,140,169,46,51,22,9,3)(215,216,217);(212,213,214)Polycyclic presentation:
2,7,3^2,211|c^a=c^2,d^a=d^2,e^a=e^210,e^b=e^171177 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26586, 42);