Finite Group DatabasebetaFinite Group Databasebeta| order | 18150 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C5×C11 |
| unambiguous semidirect product | C11⋊(C11⋊((C5⋊D5)⋊C3)) |
| Fitting, socle, derived, Hall | C52×C112 ⋊ C2×C3 |
| Hall | C112×C52⋊C3 ⋊ C2 |
| Hall | C5⋊(C5⋊(C11⋊D11)) ⋊ C3 |
| Sylow | C112 ⋊ (C5⋊D5)⋊C3 |
| Hall | C52⋊C3 ⋊ C11⋊D11 |
| Sylow | C52 ⋊ C3×C11⋊D11 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C52×C112 | C2×C3 |
| LCS | G ▷ C52×C112 | |
| UCS | C1 | |
168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 1815 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,4)(3,13,9,6,7,15)(5,10,14,11,12,8)(17,18)(19,20)(21,22)(23,24)(25,26)(28,29)(30,31)(32,33)(34,35)(36,37);(1,3,8,14,6)(2,5,7,13,11)(4,9,10,12,15)(27,28,30,32,34,36,37,35,33,31,29);(16,17,19,21,23,25,26,24,22,20,18)Polycyclic presentation:
2,3,5^2,11^2|c^a=c^4,d^a=d^4,e^a=e^10,f^a=f^10,c^b=cd,d^b=c^2d^3670 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18150, 175);