Finite Group DatabasebetaFinite Group Databasebeta| order | 19782 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#20268645 |
| Fitting, socle, derived | C3×C7×C157 ⋊ C2×C3 |
| Hall | C3×#15693073 ⋊ C2 |
| Hall | C7×C157 ⋊ C3×D3 |
| Sylow | C157 ⋊ D3×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C3⋊(D157⋊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:
(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,41)(12,26,22,20,16,13)(19,46,47,30,31,64)(21,44,45,33,29,38)(23,25,43,35,36,39)(32,69,70,49,50,93)(34,67,68,52,48,61)(37,42,66,54,55,62)(40,82,80,58,59,57)(51,98,99,72,73,124)(53,96,97,75,71,90)(56,65,95,77,78,91)(60,107,108,83,84,86)(63,113,105,87,88,81)(74,100,121,76,101,127)(79,94,126,102,103,122)(85,131,132,109,110,117)(89,135,133,114,115,112)(92,140,129,118,119,106)(104,125,144,111,128,148)(116,151,152,136,137,139)(120,154,149,141,142,134)(123,155,138,145,146,130)(143,156,157,150,147,153)(159,160,161)(162,164,163)(166,167);(1,2,5,11,19,32,51,74,75,52,33,20,4,8,13,23,37,56,79,104,109,83,58,36,10,16,24,38,57,81,106,130,136,114,87,55,18,29,39,41,61,86,112,134,150,141,118,78,31,48,59,62,64,90,117,139,153,156,145,103,50,71,84,88,91,93,121,144,155,154,151,128,73,100,110,115,119,122,124,99,126,140,135,131,101,98,125,137,142,146,148,127,70,95,113,107,96,69,94,123,147,157,152,132,97,47,66,82,67,46,65,92,120,143,149,133,108,68,28,43,44,27,42,63,89,116,138,129,105,80,45,15,26,14,25,40,60,85,111,102,77,54,35,22,7,6,12,21,34,53,76,72,49,30,17,9,3)(158,159,160,162,161,163,164)(165,166,167)Polycyclic presentation:
2,3^2,7,157|c^a=c^2,e^a=e^156,d^b=d^2,e^b=e^12561 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19782, 51);