Finite Group DatabasebetaFinite Group Databasebeta| order | 26973 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C37 · Schur multiplier (H2) = C9 |
| direct product | C37×#10624538 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C37 | #6879 |
| commutator, Frattini, cosocle | C92 | C32×C37 |
| prime radical | C37×#6879 | C3 |
| LCS | G ▷ C92 ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C37 ◁ C32×C37 ◁ C3×C9×C37 ◁ C92×C37 ◁ G | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 118 |
| transitive degree | μt(G) = 2997 |
A minimal faithful permutation representation — 2 generators on 118 points:
(1,2,6)(3,11,18)(4,9,19)(5,10,22)(7,23,13)(8,27,41)(12,35,43)(14,34,46)(15,30,47)(16,28,20)(17,31,21)(24,54,63)(25,51,38)(26,44,39)(29,49,68)(32,50,36)(33,52,66)(37,62,71)(40,57,45)(42,59,48)(53,67,58)(55,70,65)(56,73,78)(60,69,64)(61,72,80)(74,79,76)(75,81,77)(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);(1,3,12,15,38,63,42,67,79,5,14,37,41,66,78,4,13,36,17,40,65,68,80,81,16,39,64)(2,7,24,29,34,60,9,25,55,10,26,56,30,57,76,28,33,35,31,58,77,8,11,32,59,61,62)(6,18,43,47,51,54,48,53,74,22,46,71,27,52,73,19,23,50,21,45,70,49,72,75,20,44,69)3293 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26973, 98);