Finite Group DatabasebetaFinite Group Databasebeta| order | 21875 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C53×C7 · Schur multiplier (H2) = C5 |
| direct product | C7×#15691109 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C52×C7 | C53 |
| commutator, Frattini, cosocle | C52 | C53×C7 |
| prime radical | C53×C7 | C52 |
| LCS | G ▷ C52 ▷ C1 | |
| UCS | C1 ◁ C52×C7 ◁ G | |
234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 157 |
| transitive degree | μt(G) = 4375 |
A minimal faithful permutation representation — 3 generators on 157 points:
(2,6,9,16,21)(3,10,7,12,23)(8,19,15,17,20)(11,18,25,22,24)(26,27,31,41,61)(28,35,51,80,114)(29,33,43,63,91)(30,34,44,64,92)(32,45,71,106,89)(36,47,72,99,124)(37,53,66,100,126)(38,48,68,96,115)(39,49,69,62,93)(40,50,70,97,116)(42,65,98,59,78)(46,73,95,117,58)(52,67,94,118,88)(54,81,103,120,136)(55,76,107,119,137)(56,74,102,125,144)(57,77,104,122,133)(60,79,105,123,134)(75,101,121,135,87)(82,110,128,141,147)(83,108,130,140,148)(84,111,127,139,149)(85,109,129,138,150)(86,112,131,142,145)(90,113,132,143,146)(151,152,153,154,155,156,157);(1,2,7,17,25,13,16,10,20,24,4,6,3,8,18,14,21,23,19,22,5,9,12,15,11)(26,28,36,54,82,30,29,37,55,83,40,39,38,56,84,60,59,58,57,85,90,89,88,87,86)(27,32,46,74,108,34,35,52,77,111,50,33,47,75,109,79,49,53,81,112,113,78,48,76,110)(31,42,66,101,127,44,45,68,103,129,70,51,73,107,131,105,43,67,102,128,132,69,72,104,130)(41,62,94,119,138,64,65,99,125,142,97,71,100,122,141,123,80,96,121,140,143,63,95,120,139)(61,91,115,133,145,92,93,117,135,147,116,98,118,136,148,134,106,124,137,149,146,114,126,144,150);(1,3,11,6,15,4,12,24,9,20,5,10,22,16,19,13,23,25,21,17,14,7,18,2,8)(26,29,38,57,86,30,39,58,87,82,40,59,88,54,83,60,89,36,55,84,90,28,37,56,85)(27,33,48,77,112,34,49,46,75,110,50,78,52,81,108,79,32,47,76,111,113,35,53,74,109)(31,43,68,104,131,44,69,73,101,128,70,42,67,103,130,105,45,72,107,127,132,51,66,102,129)(41,63,96,122,142,64,62,95,121,141,97,65,94,120,140,123,71,99,119,139,143,80,100,125,138)(61,91,115,133,145,92,93,117,135,147,116,98,118,136,148,134,106,124,137,149,146,114,126,144,150)1715 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21875, 52);