Finite Group DatabasebetaFinite Group Databasebeta| order | 19110 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C5⋊#16838419 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C5×C13×C49 | C2×C3 |
| Frattini | C7 | C5⋊#10587722 |
| socle | C5×C7×C13 | AGL(1,𝔽7) |
| LCS | G ▷ C5×C13×C49 | |
| UCS | C1 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 3185 |
A minimal faithful permutation representation — 2 generators on 67 points:
(2,6,7,3,4,8)(5,12,11,9,10,13)(15,23,25,17,18,26)(16,35,37,21,22,38)(19,47,34,28,29,56)(20,52,53,32,33,62)(24,31,30,36,41,49)(27,50,59,45,46,60)(39,51,54,57,40,43)(42,55,44,58,48,61)(64,65)(66,67);(1,2,5,4,8,11,10,12,13,7,6,9,3)(14,59,31,54,53,40,42,21,57,19,44,49,23,24,35,34,55,26,46,45,17,37,20,33,47,29,52,32,38,15,27,50,25,48,56,22,36,18,30,61,28,39,16,58,51,62,43,41,60)(63,64,66,67,65)Polycyclic presentation:
2,3,5,7,13,7|d^7=f^6,c^a=c^4,d^a=d^6f,e^a=e^12,f^a=f^6,d^b=d^2f^3,e^b=e^3,f^b=f^2542 conjugacy classes — view the full character table.
| maximal subgroup of | #16838413⋊D5, #16838417⋊D5, D5×#16838419, … |
| maximal normal subgroup of | #16838413⋊D5, #16838417⋊D5, D5×#16838419, … |
| maximal quotient of | #50303361, C2×C5⋊#16838419 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19110, 47);