Finite Group DatabasebetaFinite Group Databasebeta| order | 26875 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C25×C43 · Schur multiplier (H2) = C1 |
| direct product | C43×C125⋊C5 |
| unambiguous semidirect product | C125⋊(C5×C43) |
| unambiguous semidirect product | (C43×C125)⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center | C25×C43 | C52 |
| commutator | C5 | C5×C25×C43 |
| Frattini, cosocle | C25 | C52×C43 |
| socle | C5×C43 | C5×C25 |
| prime radical | C52×C43 | C25 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C25×C43 ◁ G | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 168 |
| transitive degree | μt(G) = 5375 |
A minimal faithful permutation representation — 2 generators on 168 points:
(1,2,5,12,26,3,7,14,28,48,9,19,32,51,72,22,39,56,75,94,42,64,80,97,112,4,8,15,29,47,10,20,33,52,71,23,40,57,76,93,43,65,81,98,111,67,88,102,115,122,11,21,34,27,49,24,41,58,50,73,44,66,82,74,95,68,89,103,96,113,90,108,118,114,123,25,17,13,30,53,45,37,31,54,77,69,62,55,78,99,91,86,79,100,116,109,107,101,117,124,46,6,16,35,59,70,18,36,60,83,92,38,61,84,104,110,63,85,105,119,121,87,106,120,125)(126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168);(2,6,17,21,8)(5,13,15,16,34)(7,18,37,41,20)(12,27,35,29,30)(14,31,33,36,58)(19,38,62,66,40)(26,47,49,53,59)(28,50,60,52,54)(32,55,57,61,82)(39,63,86,89,65)(48,71,73,77,83)(51,74,84,76,78)(56,79,81,85,103)(64,87,107,108,88)(72,93,95,99,104)(75,96,105,98,100)(80,101,102,106,118)(94,111,113,116,119)(97,114,120,115,117)(112,122,123,124,125)6235 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26875, 6);