Finite Group DatabasebetaFinite Group Databasebeta| order | 26973 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C92×C37 · Schur multiplier (H2) = C32 |
| direct product | C37×#10624468 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C9×C37 | H(3,𝔽3) |
| commutator | C32 | C92×C37 |
| Frattini, cosocle | C32×C9 | C32×C37 |
| socle | C32×C37 | C32⋊C9 |
| prime radical | C33×C37 | C3×C9 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C9×C37 ◁ C32×C9×C37 ◁ G | |
164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 8991 |
A minimal faithful permutation representation — 2 generators on 127 points:
(1,2,4,3,5,7,6,8,9)(10,11,15)(12,20,27)(13,18,29)(14,19,28)(16,30,23)(17,34,26)(21,40,48)(22,39,49)(24,36,51)(25,37,50)(31,55,67)(32,54,44)(33,53,45)(35,59,47)(38,52,42)(41,64,71)(43,63,72)(46,61,73)(56,77,84)(57,76,68)(58,75,69)(60,80,70)(62,74,66)(65,82,86)(78,89,90)(79,88,85)(81,87,83)(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);(10,12,21,13,22,41,24,43,65,14,23,42,25,44,66,46,68,83,26,45,67,47,69,84,70,85,90)(11,16,31,18,32,56,36,57,78,19,33,40,37,58,64,61,79,82,17,20,38,35,39,62,60,63,81)(15,27,48,29,49,71,51,72,86,28,30,52,50,54,74,73,76,87,34,53,55,59,75,77,80,88,89)5661 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26973, 28);