Finite Group DatabasebetaFinite Group Databasebeta| order | 19050 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | C5⋊(C5⋊(D127⋊C3)) |
| unambiguous semidirect product | (C127⋊C3)⋊(C5⋊D5) |
| Fitting, socle, derived, Hall | C52×C127 ⋊ C2×C3 |
| Hall | C52×C127⋊C3 ⋊ C2 |
| Hall | C5⋊D635 ⋊ C3 |
| Hall | C127⋊C3 ⋊ C5⋊D5 |
| Sylow | C127 ⋊ C3×C5⋊D5 |
| Sylow | C52 ⋊ D127⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C52×C127 | C2×C3 |
| LCS | G ▷ C52×C127 | |
| UCS | C1 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 3175 |
A minimal faithful permutation representation — 3 generators on 137 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)(32,69,70,49,50,92)(34,67,68,52,48,61)(37,42,66,54,55,62)(40,81,79,58,59,57)(51,97,98,72,73,118)(53,95,96,74,71,89)(56,65,94,76,77,90)(60,105,106,82,83,85)(63,111,103,86,87,80)(75,120,121,100,99,116)(78,93,119,102,101,117)(84,123,124,107,108,114)(88,127,126,112,113,110)(91,125,122,109,115,104)(129,130)(131,132)(134,135)(136,137);(1,2,5,11,19,32,51,73,99,108,113,115,117,118,102,76,54,35,22,7,6,12,21,34,53,75,101,50,71,83,87,90,92,116,122,103,79,45,15,26,14,25,40,60,84,109,77,31,48,59,62,64,89,114,126,106,68,28,43,44,27,42,63,88,112,86,55,18,29,39,41,61,85,110,124,96,47,66,81,67,46,65,91,107,82,58,36,10,16,24,38,57,80,104,121,70,94,111,105,95,69,93,100,74,52,33,20,4,8,13,23,37,56,78,98,119,125,127,123,120,97,72,49,30,17,9,3)(133,134,136,137,135);(128,129,131,132,130)Polycyclic presentation:
2,3,5^2,127|c^a=c^4,d^a=d^4,e^a=e^126,e^b=e^107567 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19050, 17);