Finite Group DatabasebetaFinite Group Databasebeta| order | 25506 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#22555183 |
| Fitting, socle, derived | C3×C13×C109 ⋊ C2×C3 |
| Hall | C3×#16862572 ⋊ C2 |
| Hall | C13×C109 ⋊ C3×D3 |
| Sylow | C109 ⋊ C3⋊(D13⋊C3) |
| Sylow | C13 ⋊ D3×C109⋊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:
(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,35)(18,27,32)(21,30,29)(23,33,26)(25,37,51)(28,40,48)(31,43,45)(34,46,42)(36,49,39)(38,53,69)(41,55,66)(44,58,63)(47,61,60)(50,64,57)(52,67,54)(56,70,83)(59,72,80)(62,75,77)(65,78,74)(68,81,71)(73,84,94)(76,86,91)(79,89,88)(82,92,85)(87,95,102)(90,97,99)(93,100,96)(98,103,107)(101,105,104)(106,108,109)(111,115,116,112,113,117)(114,121,120,118,119,122)(124,125);(1,2,4,8,15,25,38,51,66,80,91,99,104,103,86,58,27,7,13,21,31,44,59,73,81,49,22,32,45,60,74,85,84,55,24,36,50,65,79,90,98,100,78,46,20,5,9,16,26,39,54,53,69,83,94,102,107,109,108,97,75,43,17,3,6,11,18,28,41,56,67,35,48,63,77,88,96,95,72,40,14,23,34,47,62,76,87,92,64,33,12,19,29,42,57,71,70,37,52,68,82,93,101,106,105,89,61,30,10)(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,d^b=d^3,e^b=e^63771 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(25506, 59);