Finite Group DatabasebetaFinite Group Databasebeta| order | 21875 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C53×C7 · Schur multiplier (H2) = C53 |
| direct product | C7×#15691122 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C25 | H(3,𝔽5) |
| commutator, Frattini, cosocle | C52 | C53×C7 |
| socle | C5×C7 | C5×H(3,𝔽5) |
| prime radical | C7×H(3,𝔽5)⋊C5 | C5 |
| LCS | G ▷ C52 ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C7×C25 ◁ C5×C7×C25 ◁ G | |
294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 132 |
| transitive degree | μt(G) = 875 |
A minimal faithful permutation representation — 3 generators on 132 points:
(1,2,5,12,26)(3,7,14,28,47)(4,6,15,27,48)(8,20,41,66,46)(9,18,32,50,71)(10,17,33,49,72)(11,16,34,29,52)(13,30,53,25,21)(19,39,65,89,70)(22,37,56,74,93)(23,36,57,73,94)(24,35,58,51,76)(31,54,77,45,40)(38,63,88,108,92)(42,61,80,96,111)(43,60,81,95,112)(44,59,82,75,98)(55,78,99,69,64)(62,86,107,120,110)(67,84,102,113,122)(68,83,103,97,115)(79,100,116,91,87)(85,105,119,125,121)(90,104,118,114,123)(101,117,124,109,106)(126,127,128,129,130,131,132);(2,6,16,21,8)(5,13,15,20,34)(7,17,35,40,19)(12,27,29,30,41)(14,31,33,39,58)(18,36,59,64,38)(28,49,51,54,65)(32,55,57,63,82)(37,60,83,87,62)(50,73,75,78,88)(56,79,81,86,103)(61,84,104,106,85)(74,95,97,100,107)(80,101,102,105,118)(96,113,114,117,119);(1,3,9,22,42,4,10,23,43,67,11,24,44,68,90,25,45,69,91,109,46,70,92,110,121)(2,7,18,37,61,6,17,36,60,84,16,35,59,83,104,21,40,64,87,106,8,19,38,62,85)(5,14,32,56,80,15,33,57,81,102,34,58,82,103,118,13,31,55,79,101,20,39,63,86,105)(12,28,50,74,96,27,49,73,95,113,29,51,75,97,114,30,54,78,100,117,41,65,88,107,119)(26,47,71,93,111,48,72,94,112,122,52,76,98,115,123,53,77,99,116,124,66,89,108,120,125)1715 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21875, 65);