Finite Group DatabasebetaFinite Group Databasebeta| order | 21875 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C7×C125 · Schur multiplier (H2) = C52 |
| direct product | C7×C52⋊C125 |
| unambiguous semidirect product | C52⋊(C7×C125) |
| unambiguous semidirect product | (C52×C7)⋊C125 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C7×C25 | C52 |
| commutator | C5 | C5×C7×C125 |
| Frattini, cosocle | C5×C25 | C52×C7 |
| socle | C52×C7 | C5×C25 |
| prime radical | C53×C7 | C25 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C5×C7×C25 ◁ G | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 157 |
| transitive degree | μt(G) = 4375 |
A minimal faithful permutation representation — 2 generators on 157 points:
(1,2,4,8,15)(3,6,10,17,22)(5,11,20,25,21)(7,13,19,16,23)(9,18,24,14,12)(26,27,30,36,46,28,31,37,47,61,33,39,49,63,79,42,52,66,82,98,56,70,86,102,116,29,32,38,48,62,34,40,50,64,80,43,53,67,83,99,57,71,87,103,117,75,91,107,121,131,35,41,51,65,81,44,54,68,84,100,58,72,88,104,118,76,92,108,122,132,95,111,125,135,141,45,55,69,85,101,59,73,89,105,119,77,93,109,123,133,96,112,126,136,142,114,128,138,144,147,60,74,90,106,120,78,94,110,124,134,97,113,127,137,143,115,129,139,145,148,130,140,146,149,150)(151,152,153,154,155,156,157);(2,5,12,13,6)(4,9,10,11,19)(8,16,20,17,18)(15,22,23,24,25)5075 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21875, 17);