Finite Group DatabasebetaFinite Group Databasebeta| order | 26973 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C9×C37 · Schur multiplier (H2) = C35 |
| direct product | C37×#10624918 |
| subgroup | G/H | |
|---|---|---|
| center | C9×C37 | C34 |
| commutator | C3 | C33×C9×C37 |
| Frattini, cosocle | C9 | C34×C37 |
| socle | C3×C37 | C33×C9 |
| prime radical | C3×H(3,𝔽3)×C37 | C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C9×C37 ◁ G | |
828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 118 |
| transitive degree | μt(G) = 2997 |
A minimal faithful permutation representation — 4 generators on 118 points:
(1,2,6,4,9,20,15,27,42,5,10,19,16,28,41,35,51,64,17,8,25,36,26,49,57,50,70)(3,7,18,13,23,40,33,47,63,14,24,39,34,48,62,56,69,77,12,22,38,32,46,61,55,68,76)(11,21,37,30,45,59,53,67,74,29,43,58,52,65,73,71,78,80,31,44,60,54,66,75,72,79,81)(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);(1,3,11)(2,7,21)(4,13,30)(5,14,29)(6,18,37)(8,22,44)(9,23,45)(10,24,43)(12,31,17)(15,33,53)(16,34,52)(19,39,58)(20,40,59)(25,38,60)(26,46,66)(27,47,67)(28,48,65)(32,54,36)(35,56,71)(41,62,73)(42,63,74)(49,61,75)(50,68,79)(51,69,78)(55,72,57)(64,77,80)(70,76,81);(3,12,14)(7,22,24)(11,29,31)(13,32,34)(18,38,39)(21,43,44)(23,46,48)(30,52,54)(33,55,56)(37,58,60)(40,61,62)(45,65,66)(47,68,69)(53,71,72)(59,73,75)(63,76,77)(67,78,79)(74,80,81);(2,8,10)(6,19,25)(7,22,24)(9,26,28)(18,39,38)(20,41,49)(21,44,43)(23,46,48)(27,50,51)(37,58,60)(40,62,61)(42,64,70)(45,66,65)(47,68,69)(59,73,75)(63,77,76)(67,79,78)(74,80,81)9213 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26973, 478);