Finite Group DatabasebetaFinite Group Databasebeta| order | 20178 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊((C19⋊C3)⋊D59) |
| unambiguous semidirect product | (C19⋊C3)⋊D177 |
| unambiguous semidirect product | C59⋊(C3⋊(D19⋊C3)) |
| unambiguous semidirect product | D3363⋊C3 |
| Fitting, socle, derived | C3×C19×C59 ⋊ C2×C3 |
| Hall | C3×C19⋊C3×C59 ⋊ C2 |
| Hall | C19×C59 ⋊ C3×D3 |
| Hall | C3×C19⋊C3 ⋊ D59 |
| Sylow | C59 ⋊ C3⋊(D19⋊C3) |
| Sylow | C19 ⋊ C3×D177 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C19×C59 | C2×C3 |
| LCS | G ▷ C3×C19×C59 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 3363 |
A minimal faithful permutation representation — 2 generators on 81 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)(80,81);(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)(79,80,81)Polycyclic presentation:
2,3^2,19,59|c^a=c^2,d^a=d^18,e^a=e^58,d^b=d^7801 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20178, 22);