Finite Group DatabasebetaFinite Group Databasebeta| order | 26973 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C92×C37 · Schur multiplier (H2) = C3×C9 |
| direct product | C37×#10624681 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C27×C37 | C32 |
| commutator | C3 | C3×C92×C37 |
| Frattini, cosocle | C3×C9 | C33×C37 |
| socle | C32×C37 | C32×C9 |
| prime radical | C33×C37 | C3×C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C27×C37 ◁ G | |
290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 8991 |
A minimal faithful permutation representation — 3 generators on 127 points:
(10,11,15,13,17,27,23,33,48,14,18,28,24,34,49,42,57,71,25,35,50,43,58,72,64,78,86)(12,16,26,21,31,46,40,55,69,22,32,47,41,56,70,63,77,85,20,30,45,39,54,68,62,76,84)(19,29,44,37,52,66,60,74,82,36,51,65,59,73,81,79,87,89,38,53,67,61,75,83,80,88,90)(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);(1,2,4,3,5,7,6,8,9)(10,12,19)(11,16,29)(13,21,37)(14,22,36)(15,26,44)(17,31,52)(18,32,51)(20,38,25)(23,40,60)(24,41,59)(27,46,66)(28,47,65)(30,53,35)(33,55,74)(34,56,73)(39,61,43)(42,63,79)(45,67,50)(48,69,82)(49,70,81)(54,75,58)(57,77,87)(62,80,64)(68,83,72)(71,85,89)(76,88,78)(84,90,86);(12,20,22)(16,30,32)(19,36,38)(21,39,41)(26,45,47)(29,51,53)(31,54,56)(37,59,61)(40,62,63)(44,65,67)(46,68,70)(52,73,75)(55,76,77)(60,79,80)(66,81,83)(69,84,85)(74,87,88)(82,89,90)10989 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26973, 241);