Finite Group DatabasebetaFinite Group Databasebeta| order | 21875 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C52×C7 · Schur multiplier (H2) = C1 |
| direct product | C7×#15691065 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C52×C7 | H(3,𝔽5) |
| commutator, Frattini, cosocle | C53 | C52×C7 |
| prime radical | C53×C7 | C52 |
| LCS | G ▷ C53 ▷ C52 ▷ C1 | |
| UCS | C1 ◁ C52×C7 ◁ C53×C7 ◁ G | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 157 |
| transitive degree | μt(G) = 4375 |
A minimal faithful permutation representation — 2 generators on 157 points:
(1,2,6,18,24,16,21,15,20,19,4,9,3,11,22,17,8,13,7,12,5,10,14,23,25)(26,27,30,38,56,29,33,41,59,85,37,47,64,58,89,55,46,40,63,95,83,32,45,71,101)(28,34,48,72,84,36,31,42,57,86,54,44,66,88,115,82,70,39,60,90,112,50,62,92,119)(35,51,75,105,113,53,49,61,93,114,81,43,67,87,116,111,69,73,102,118,132,77,65,91,120)(52,78,97,121,133,80,76,103,129,134,110,74,96,122,135,131,68,98,117,136,144,100,94,123,138)(79,108,125,141,145,109,107,127,139,146,130,106,124,142,147,143,104,126,140,148,150,99,128,137,149)(151,152,153,154,155,156,157);(1,3,12,9,18,4,14,24,10,11,5,15,22,21,23,16,13,25,8,20,17,6,19,2,7)(26,28,35,52,79,29,36,53,80,109,37,54,81,110,130,55,82,111,131,143,83,112,132,144,150)(27,31,43,68,99,33,44,69,100,108,47,70,77,78,107,46,50,51,76,106,32,34,49,74,104)(30,39,61,94,124,41,62,67,97,126,64,48,73,103,128,40,42,65,96,125,45,66,75,98,127)(38,57,87,117,137,59,88,102,123,141,58,60,91,121,139,63,92,105,129,142,71,72,93,122,140)(56,84,113,133,145,85,86,114,134,146,89,115,116,135,147,95,90,118,136,148,101,119,120,138,149)1043 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21875, 8);