Finite Group DatabasebetaFinite Group Databasebeta| order | 13300 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | C7⋊(C19⋊#1044) |
| unambiguous semidirect product | C19⋊(C7⋊#1044) |
| Fitting, socle, derived, Hall | C52×C7×C19 ⋊ C4 |
| Hall | C52×C19 ⋊ Dic7 |
| Hall | C52×C7 ⋊ Dic19 |
| Hall | C7×C19 ⋊ #1044 |
| Sylow | C52 ⋊ Dic133 |
| Sylow | C19 ⋊ C7⋊#1044 |
| Sylow | C7 ⋊ C19⋊#1044 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C52×C7×C19 | C4 |
| prime radical, cosocle | C7×C19×C5⋊D5 | C2 |
| LCS | G ▷ C52×C7×C19 | |
| UCS | C1 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 1330 |
A minimal faithful permutation representation — 2 generators on 36 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(21,22)(23,24)(25,26)(28,29,30,31)(33,34,35,36);(1,2,4,6,8,10,12,14,16,18,19,17,15,13,11,9,7,5,3)(20,21,23,25,26,24,22)(27,28,29,31,30)(32,33,36,34,35)Polycyclic presentation:
2^2,5^2,7,19|a^2=b,c^a=c^2,d^a=d^3,e^a=e^6,f^a=f^18,c^b=c^4,d^b=d^4934 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13300, 59);