Finite Group DatabasebetaFinite Group Databasebeta| order | 26166 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C89⋊#64424 |
| Fitting, socle, derived, Hall | C72×C89 ⋊ C2×C3 |
| Hall | C89×#3861 ⋊ C2 |
| Hall | C7⋊D623 ⋊ C3 |
| Hall | #3861 ⋊ D89 |
| Sylow | C89 ⋊ #64424 |
| Sylow | C72 ⋊ C3×D89 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C72×C89 | C2×C3 |
| LCS | G ▷ C72×C89 | |
| UCS | C1 | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 103 |
| transitive degree | μt(G) = 4361 |
A minimal faithful permutation representation — 3 generators on 103 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)(91,94,95,92,93,96)(98,101,102,99,100,103);(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)(97,98,100,103,102,101,99);(90,91,93,96,95,94,92)Polycyclic presentation:
2,3,7^2,89|c^a=c^6,d^a=d^6,e^a=e^88,c^b=c^2,d^b=d^2850 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26166, 25);