Finite Group DatabasebetaFinite Group Databasebeta| order | 26973 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C9×C37 · Schur multiplier (H2) = C9 |
| direct product | C37×#10624713 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C9×C37 | C33 |
| commutator | C32 | C32×C9×C37 |
| Frattini, cosocle | C3×C9 | C33×C37 |
| socle | C32×C37 | C32×C9 |
| prime radical | C33×C37 | C3×C9 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C3×C9×C37 ◁ G | |
172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 145 |
| transitive degree | μt(G) = 8991 |
A minimal faithful permutation representation — 3 generators on 145 points:
(1,2,5)(3,7,13)(4,8,12)(6,14,11)(9,16,21)(10,17,20)(15,22,19)(18,24,26)(23,27,25)(28,29,33,31,35,44,40,49,63,32,36,45,41,50,64,58,72,86,42,51,65,59,73,87,79,93,102)(30,34,43,39,48,62,57,71,85,38,47,61,56,70,84,78,92,101,55,69,83,77,91,100,96,105,108)(37,46,60,53,67,81,75,89,98,54,68,82,76,90,99,95,104,107,52,66,80,74,88,97,94,103,106)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145);(2,6,8)(5,12,14)(7,15,17)(13,20,22)(16,23,24)(21,26,27)(28,30,37)(29,34,46)(31,39,53)(32,38,54)(33,43,60)(35,48,67)(36,47,68)(40,57,75)(41,56,76)(42,55,52)(44,62,81)(45,61,82)(49,71,89)(50,70,90)(51,69,66)(58,78,95)(59,77,74)(63,85,98)(64,84,99)(65,83,80)(72,92,104)(73,91,88)(79,96,94)(86,101,107)(87,100,97)(93,105,103)(102,108,106);(1,3,9,4,10,18,11,19,25)(2,7,16,8,17,24,6,15,23)(5,13,21,12,20,26,14,22,27)(30,38,55)(34,47,69)(37,52,54)(39,56,77)(43,61,83)(46,66,68)(48,70,91)(53,74,76)(57,78,96)(60,80,82)(62,84,100)(67,88,90)(71,92,105)(75,94,95)(81,97,99)(85,101,108)(89,103,104)(98,106,107)5661 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26973, 273);