Finite Group DatabasebetaFinite Group Databasebeta| order | 21875 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C53×C7 · Schur multiplier (H2) = C53 |
| direct product | C7×#15691127 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C5×C7 | C5×H(3,𝔽5) |
| commutator, Frattini, cosocle | C52 | C53×C7 |
| prime radical | C7×H(3,𝔽5)⋊C5 | C5 |
| LCS | G ▷ C52 ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C5×C7 ◁ C5×C7×C25 ◁ G | |
278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 132 |
| transitive degree | μt(G) = 875 |
A minimal faithful permutation representation — 3 generators on 132 points:
(1,2,5,12,28)(3,7,14,30,54)(4,6,15,29,55)(8,21,45,73,53)(9,17,36,56,80)(10,19,35,58,81)(11,16,37,31,59)(13,32,60,27,22)(18,40,57,83,23)(20,34,63,89,74)(24,39,66,82,99)(25,41,68,84,100)(26,43,65,86,101)(33,61,87,46,38)(42,62,85,104,47)(44,64,90,109,49)(48,67,92,102,113)(50,69,95,103,114)(51,70,88,105,115)(52,72,93,107,116)(71,91,106,120,75)(76,94,111,117,122)(77,96,112,118,123)(78,97,108,119,124)(79,98,110,121,125)(126,127,128,129,130,131,132);(2,6,16,22,8)(3,9,23,46,74)(5,13,15,21,37)(7,18,20,17,38)(10,25,49,24,47)(12,29,31,32,45)(14,34,33,40,36)(19,42,39,44,41)(26,51,48,75,50)(30,57,63,56,61)(43,71,70,69,67)(52,79,78,77,76)(54,80,83,87,89)(58,85,82,90,84)(65,92,95,88,91)(81,100,109,99,104)(86,106,105,103,102)(93,112,110,111,108)(101,115,113,120,114)(116,125,124,123,122);(1,3,10,26,52,4,9,24,48,76,11,23,25,50,77,27,46,47,51,78,53,74,49,75,79)(2,7,19,43,72,6,17,39,67,94,16,18,41,69,96,22,38,42,70,97,8,20,44,71,98)(5,14,35,65,93,15,36,66,92,111,37,40,68,95,112,13,33,62,88,108,21,34,64,91,110)(12,30,58,86,107,29,56,82,102,117,31,57,84,103,118,32,61,85,105,119,45,63,90,106,121)(28,54,81,101,116,55,80,99,113,122,59,83,100,114,123,60,87,104,115,124,73,89,109,120,125)1043 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21875, 70);