Finite Group DatabasebetaFinite Group Databasebeta| order | 20874 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C71⋊#64424 |
| Fitting, socle, derived, Hall | C72×C71 ⋊ C2×C3 |
| Hall | C71×#3861 ⋊ C2 |
| Hall | C7⋊D497 ⋊ C3 |
| Hall | #3861 ⋊ D71 |
| Sylow | C71 ⋊ #64424 |
| Sylow | C72 ⋊ C3×D71 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C72×C71 | C2×C3 |
| LCS | G ▷ C72×C71 | |
| UCS | C1 | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3479 |
A minimal faithful permutation representation — 3 generators on 85 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)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(73,76,77,74,75,78)(80,83,84,81,82,85);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(79,80,82,85,84,83,81);(72,73,75,78,77,76,74)Polycyclic presentation:
2,3,7^2,71|c^a=c^6,d^a=d^6,e^a=e^70,c^b=c^2,d^b=d^2679 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20874, 41);