Finite Group DatabasebetaFinite Group Databasebeta| order | 26973 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C9×C37 · Schur multiplier (H2) = C9 |
| direct product | C37×#10624712 |
| 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)(28,29,33,31,35,45,41,51,66,32,36,46,42,52,67,60,75,89,43,53,68,61,76,90,82,96,104)(30,34,44,39,49,64,58,73,87,40,50,65,59,74,88,81,95,103,38,48,63,57,72,86,80,94,102)(37,47,62,55,70,84,78,92,100,54,69,83,77,91,99,97,105,107,56,71,85,79,93,101,98,106,108)(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)(29,34,47)(31,39,55)(32,40,54)(33,44,62)(35,49,70)(36,50,69)(38,56,43)(41,58,78)(42,59,77)(45,64,84)(46,65,83)(48,71,53)(51,73,92)(52,74,91)(57,79,61)(60,81,97)(63,85,68)(66,87,100)(67,88,99)(72,93,76)(75,95,105)(80,98,82)(86,101,90)(89,103,107)(94,106,96)(102,108,104);(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)(30,38,40)(34,48,50)(37,54,56)(39,57,59)(44,63,65)(47,69,71)(49,72,74)(55,77,79)(58,80,81)(62,83,85)(64,86,88)(70,91,93)(73,94,95)(78,97,98)(84,99,101)(87,102,103)(92,105,106)(100,107,108)5661 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26973, 272);