Finite Group DatabasebetaFinite Group Databasebeta| order | 25506 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#22555187 |
| Fitting, socle, derived | C3×C13×C109 ⋊ C2×C3 |
| Hall | C3×#16862572 ⋊ C2 |
| Hall | C13×C109 ⋊ C3×D3 |
| Sylow | C109 ⋊ C3⋊(D13⋊C3) |
| Sylow | C13 ⋊ C3⋊(D109⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C13×C109 | C2×C3 |
| LCS | G ▷ C3×C13×C109 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 4251 |
A minimal faithful permutation representation — 2 generators on 125 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)(32,69,70,49,50,91)(34,67,68,52,48,61)(37,42,66,54,55,62)(40,80,78,58,59,57)(51,96,97,72,73,109)(53,94,95,74,71,88)(56,65,93,75,76,89)(60,101,102,81,82,84)(63,105,99,85,86,79)(77,92,103,98,83,108)(87,107,100,90,106,104)(111,115,116,112,113,117)(114,121,120,118,119,122)(124,125);(1,2,5,11,19,32,51,70,93,105,101,94,69,92,81,58,36,10,16,24,38,57,79,100,102,68,28,43,44,27,42,63,87,76,31,48,59,62,64,88,98,75,54,35,22,7,6,12,21,34,53,73,97,103,106,107,108,109,96,74,52,33,20,4,8,13,23,37,56,77,95,47,66,80,67,46,65,90,85,55,18,29,39,41,61,84,104,99,78,45,15,26,14,25,40,60,83,50,71,82,86,89,91,72,49,30,17,9,3)(110,111,114,113,117,120,119,121,122,116,115,118,112)(123,124,125)Polycyclic presentation:
2,3^2,13,109|c^a=c^2,d^a=d^12,e^a=e^108,d^b=d^3,e^b=e^63717 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(25506, 57);