Finite Group DatabasebetaFinite Group Databasebeta| order | 19782 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#20268649 |
| Fitting, socle, derived | C3×C7×C157 ⋊ C2×C3 |
| Hall | C3×#15693073 ⋊ C2 |
| Hall | C7×C157 ⋊ C3×D3 |
| Sylow | C157 ⋊ C3⋊AGL(1,𝔽7) |
| Sylow | C7 ⋊ D3×C157⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C7×C157 | C2×C3 |
| LCS | G ▷ C3×C7×C157 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 3297 |
A minimal faithful permutation representation — 2 generators on 167 points:
(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,35)(18,27,32)(21,30,29)(23,33,26)(25,37,51)(28,40,48)(31,43,45)(34,46,42)(36,49,39)(38,53,70)(41,56,67)(44,59,64)(47,62,61)(50,65,58)(52,68,55)(54,72,92)(57,75,89)(60,78,86)(63,81,83)(66,84,80)(69,87,77)(71,90,74)(73,94,106)(76,96,113)(79,99,110)(82,102,107)(85,105,104)(88,108,101)(91,111,98)(93,114,95)(97,118,126)(100,120,132)(103,123,129)(109,127,125)(112,130,122)(115,133,119)(116,128,117)(121,137,142)(124,139,145)(131,143,141)(134,146,138)(135,144,136)(140,150,152)(147,153,151)(148,154,149)(155,157,156)(159,162,163,160,161,164)(166,167);(1,2,4,8,15,25,38,54,73,81,43,17,3,6,11,18,28,41,57,76,97,102,59,27,7,13,21,31,44,60,79,100,121,123,78,40,14,23,34,47,63,82,103,124,140,139,99,56,24,36,50,66,85,106,126,142,152,150,120,75,37,52,69,88,109,128,92,113,132,145,137,96,53,71,91,112,131,144,114,70,89,110,129,118,72,93,115,134,147,154,133,90,51,67,86,107,94,116,135,148,155,157,146,111,68,35,48,64,83,104,125,141,151,156,153,130,87,49,22,32,45,61,80,101,122,138,149,143,108,65,33,12,19,29,42,58,77,98,119,136,127,84,46,20,5,9,16,26,39,55,74,95,117,105,62,30,10)(158,159,161,164,163,162,160)(165,166,167)Polycyclic presentation:
2,3^2,7,157|c^a=c^2,d^a=d^6,d^b=d^2,e^b=e^12636 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19782, 47);