Finite Group DatabasebetaFinite Group Databasebeta| order | 36354 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C83⋊(D73⋊C3) |
| unambiguous semidirect product | (C73⋊C3)⋊D83 |
| unambiguous semidirect product | D6059⋊C3 |
| Fitting, socle, derived, Hall | C73×C83 ⋊ C2×C3 |
| Hall | C83×C73⋊C3 ⋊ C2 |
| Hall | D6059 ⋊ C3 |
| Hall | C73⋊C3 ⋊ D83 |
| Sylow | C83 ⋊ D73⋊C3 |
| Sylow | C73 ⋊ C3×D83 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C73×C83 | C2×C3 |
| LCS | G ▷ C73×C83 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 156 |
| transitive degree | μt(G) = 6059 |
A minimal faithful permutation representation — 2 generators on 156 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)(19,46,47,30,31,60)(21,44,45,33,29,38)(23,25,43,35,36,39)(32,48,57,34,49,63)(37,42,62,50,51,58)(40,67,65,54,55,53)(52,61,68,56,64,72)(59,73,71,69,70,66)(75,76)(77,78)(79,80)(81,82)(83,84)(85,86)(87,88)(89,90)(91,92)(93,94)(95,96)(97,98)(99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156);(1,2,5,11,19,32,33,20,4,8,13,23,37,52,54,36,10,16,24,38,53,66,69,51,18,29,39,41,57,68,73,64,31,48,55,58,60,47,62,67,49,46,61,70,72,63,28,43,44,27,42,59,71,65,45,15,26,14,25,40,56,50,35,22,7,6,12,21,34,30,17,9,3)(74,75,77,79,81,83,85,87,89,91,93,95,97,99,101,103,105,107,109,111,113,115,117,119,121,123,125,127,129,131,133,135,137,139,141,143,145,147,149,151,153,155,156,154,152,150,148,146,144,142,140,138,136,134,132,130,128,126,124,122,120,118,116,114,112,110,108,106,104,102,100,98,96,94,92,90,88,86,84,82,80,78,76)Polycyclic presentation:
2,3,73,83|c^a=c^72,d^a=d^82,c^b=c^81125 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36354, 12);