Finite Group DatabasebetaFinite Group Databasebeta| order | 13452 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C19⋊C3)⋊Dic59 |
| unambiguous semidirect product | C59⋊((C19⋊C3)⋊C4) |
| unambiguous semidirect product | Dic1121⋊C3 |
| derived, Hall | C19×C59 ⋊ C3×C4 |
| Hall | Dic1121 ⋊ C3 |
| Hall | C19⋊C3×C59 ⋊ C4 |
| Sylow | C59 ⋊ (C19⋊C3)⋊C4 |
| Hall | C19⋊C3 ⋊ Dic59 |
| Sylow | C19 ⋊ C3×Dic59 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | (C19⋊C3)⋊D59 |
| commutator, hypocenter | C19×C59 | C3×C4 |
| Fitting, socle, cosocle | C2×C19×C59 | C2×C3 |
| prime radical | C2×C19⋊C3×C59 | C2 |
| LCS | G ▷ C19×C59 | |
| UCS | C1 ◁ C2 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 82 |
| transitive degree | μt(G) = 4484 |
A minimal faithful permutation representation — 2 generators on 82 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(61,65,66,62,63,67)(64,73,70,68,69,75)(71,76,78,77,74,72)(79,80,81,82);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(60,61,64,70,76,73,77,63,67,72,78,66,65,71,69,74,75,68,62)Polycyclic presentation:
2,3,19,59,2|a^2=e,c^a=c^18,d^a=d^58,c^b=c^7540 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13452, 12);