Finite Group DatabasebetaFinite Group Databasebeta| order | 26875 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C53×C43 · Schur multiplier (H2) = C52 |
| direct product | C43×#10597837 |
| subgroup | G/H | |
|---|---|---|
| center | C25×C43 | C52 |
| commutator, Frattini, cosocle | C5 | C53×C43 |
| socle | C5×C43 | C53 |
| prime radical | C43×H(3,𝔽5) | C5 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C25×C43 ◁ G | |
142 conjugacy classes of subgroups.
| minimal degree | μ(G) = 168 |
| transitive degree | μt(G) = 5375 |
A minimal faithful permutation representation — 3 generators on 168 points:
(1,2,5,12,26)(3,7,14,28,48)(4,8,15,29,47)(6,16,35,59,46)(9,19,32,51,72)(10,20,33,52,71)(11,21,34,27,49)(13,30,53,25,17)(18,36,60,83,70)(22,39,56,75,94)(23,40,57,76,93)(24,41,58,50,73)(31,54,77,45,37)(38,61,84,104,92)(42,64,80,97,112)(43,65,81,98,111)(44,66,82,74,95)(55,78,99,69,62)(63,85,105,119,110)(67,88,102,115,122)(68,89,103,96,113)(79,100,116,91,86)(87,106,120,125,121)(90,108,118,114,123)(101,117,124,109,107)(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);(1,3,9,22,42,4,10,23,43,67,11,24,44,68,90,25,45,69,91,109,46,70,92,110,121)(2,7,19,39,64,8,20,40,65,88,21,41,66,89,108,17,37,62,86,107,6,18,38,63,87)(5,14,32,56,80,15,33,57,81,102,34,58,82,103,118,13,31,55,79,101,16,36,61,85,106)(12,28,51,75,97,29,52,76,98,115,27,50,74,96,114,30,54,78,100,117,35,60,84,105,120)(26,48,72,94,112,47,71,93,111,122,49,73,95,113,123,53,77,99,116,124,59,83,104,119,125)6235 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26875, 14);