Finite Group DatabasebetaFinite Group Databasebeta| order | 26973 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C34×C37 · Schur multiplier (H2) = C36 |
| direct product | C37×#10624933 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C37 | C32×H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C34×C37 |
| prime radical | C3×H(3,𝔽3)×C37 | C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C37 ◁ C3×H(3,𝔽3)×C37 ◁ G | |
936 conjugacy classes of subgroups.
| minimal degree | μ(G) = 118 |
| transitive degree | μt(G) = 2997 |
A minimal faithful permutation representation — 4 generators on 118 points:
(1,2,6,5,10,21,18,9,29)(3,11,19,14,7,22,38,25,48)(4,8,20,17,28,45,16,27,44)(12,32,41,34,30,46,61,23,49)(13,31,42,37,24,47,36,52,66)(15,26,43,39,53,67,40,54,68)(33,57,64,60,55,69,59,50,71)(35,56,65,62,51,70,63,73,79)(58,75,78,76,74,80,77,72,81)(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);(1,3,12,18,38,61,5,14,34)(2,7,23,9,11,30,10,25,32)(4,13,33,16,36,59,17,37,60)(6,19,41,29,48,49,21,22,46)(8,24,50,27,31,55,28,52,57)(15,35,58,40,63,77,39,62,76)(20,42,64,44,66,71,45,47,69)(26,51,72,54,56,74,53,73,75)(43,65,78,68,79,81,67,70,80);(1,4,15)(2,8,26)(3,13,35)(5,17,39)(6,20,43)(7,24,51)(9,27,54)(10,28,53)(11,31,56)(12,33,58)(14,37,62)(16,40,18)(19,42,65)(21,45,67)(22,47,70)(23,50,72)(25,52,73)(29,44,68)(30,55,74)(32,57,75)(34,60,76)(36,63,38)(41,64,78)(46,69,80)(48,66,79)(49,71,81)(59,77,61);(4,16,17)(8,27,28)(13,36,37)(15,39,40)(20,44,45)(24,31,52)(26,53,54)(33,59,60)(35,62,63)(42,66,47)(43,67,68)(50,55,57)(51,73,56)(58,76,77)(64,71,69)(65,70,79)(72,75,74)(78,80,81)3885 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26973, 493);