Finite Group DatabasebetaFinite Group Databasebeta| order | 26973 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9×C37 · Schur multiplier (H2) = C3 |
| direct product | C37×#10624522 |
| subgroup | G/H | |
|---|---|---|
| center | C9×C37 | #732 |
| commutator | C3×C9 | C3×C9×C37 |
| Frattini, cosocle | C32×C9 | C32×C37 |
| socle | C3×C37 | #6868 |
| prime radical | C33×C37 | 3−1+2 |
| LCS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C9×C37 ◁ C3×C9×C37 ◁ C32×C9×C37 ◁ G | |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 118 |
| transitive degree | μt(G) = 2997 |
A minimal faithful permutation representation — 2 generators on 118 points:
(1,2,7,5,10,26,21,9,33,6,12,27,22,35,60,54,36,69,23,11,38,20,37,59,4,18,25)(3,13,39,16,8,28,50,30,62,17,34,70,51,32,58,15,47,24,49,42,56,53,31,68,19,41,57)(14,43,75,45,64,80,44,66,72,46,74,61,76,40,73,48,29,63,77,65,81,78,71,55,52,67,79)(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);(1,3,14)(2,8,29,12,32,67,11,31,66)(4,19,52)(5,16,45)(6,17,46)(7,24,55,38,62,73,27,57,80)(9,34,71,36,42,64,18,13,40)(10,30,65,35,47,43,37,41,74)(15,48,54)(20,53,78)(21,50,44)(22,51,76)(23,49,77)(25,58,81,69,28,61,33,68,75)(26,56,79,59,70,63,60,39,72)2109 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26973, 82);