Finite Group DatabasebetaFinite Group Databasebeta| order | 13300 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D133⋊D25 |
| unambiguous semidirect product | D175⋊D19 |
| unambiguous semidirect product | D475⋊D7 |
| Fitting, derived, Hall | C7×C19×C25 ⋊ C22 |
| Hall | C19×C25 ⋊ C2×D7 |
| Hall | C7×C25 ⋊ C2×D19 |
| Hall | C7×C19 ⋊ C2×D25 |
| Sylow | C25 ⋊ D7×D19 |
| Sylow | C19 ⋊ D7×D25 |
| Sylow | C7 ⋊ D19×D25 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C7×C19×C25 | C22 |
| Frattini | C5 | D35⋊D19 |
| socle | C5×C7×C19 | C2×D5 |
| LCS | G ▷ C7×C19×C25 | |
| UCS | C1 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 3325 |
A minimal faithful permutation representation — 2 generators on 51 points:
(1,2,4,6,8,10,12,14,16,18,19,17,15,13,11,9,7,5,3)(20,21)(22,23)(24,25)(28,30)(29,33)(31,36)(32,39)(34,42)(35,41)(37,38)(40,46)(43,47)(44,48)(45,50)(49,51);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(21,22)(23,24)(25,26)(27,41,51,44,30,33,47,45,36,39,42,40,38,37,46,34,32,31,50,43,29,28,48,49,35)Polycyclic presentation:
2^2,5,7,19,5|c^5=f^4,d^a=d^6,e^a=e^18,c^b=c^4f,d^b=d^6,f^b=f^4871 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13300, 20);