Finite Group DatabasebetaFinite Group Databasebeta| order | 26166 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C89⋊(D49⋊C3) |
| unambiguous semidirect product | (C49⋊C3)⋊D89 |
| unambiguous semidirect product | D4361⋊C3 |
| Fitting, derived, Hall | C49×C89 ⋊ C2×C3 |
| Hall | C89×C49⋊C3 ⋊ C2 |
| Hall | D4361 ⋊ C3 |
| Hall | C49⋊C3 ⋊ D89 |
| Sylow | C89 ⋊ D49⋊C3 |
| Sylow | C49 ⋊ C3×D89 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C49×C89 | C2×C3 |
| Frattini | C7 | ΓL(1,𝔽8)⋊D89 |
| socle | C7×C89 | AGL(1,𝔽7) |
| LCS | G ▷ C49×C89 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 138 |
| transitive degree | μt(G) = 4361 |
A minimal faithful permutation representation — 2 generators on 138 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)(86,87)(88,89)(91,99,101,93,94,102)(92,111,113,97,98,114)(95,123,110,104,105,132)(96,128,129,108,109,138)(100,107,106,112,117,125)(103,126,135,121,122,136)(115,127,130,133,116,119)(118,131,120,134,124,137);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(90,112,109,120,119,124,93,97,134,103,102,129,104,108,111,135,94,123,125,117,132,113,133,127,126,122,116,115,114,110,107,106,105,99,136,98,96,95,138,101,121,118,92,91,131,130,137,128,100)Polycyclic presentation:
2,3,7,89,7|c^7=e^6,c^a=c^6e,d^a=d^88,e^a=e^6,c^b=c^2e^3,e^b=e^2850 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26166, 12);