Finite Group DatabasebetaFinite Group Databasebeta| order | 21875 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C53×C7 · Schur multiplier (H2) = C53 |
| direct product | C7×#15691124 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C5×C7 | C5×H(3,𝔽5) |
| commutator, Frattini, cosocle | C52 | C53×C7 |
| prime radical | C5×C7×H(3,𝔽5) | C5 |
| LCS | G ▷ C52 ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C5×C7 ◁ C53×C7 ◁ G | |
296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 132 |
| transitive degree | μt(G) = 875 |
A minimal faithful permutation representation — 3 generators on 132 points:
(1,2,5,14,35,4,8,17,38,66,13,23,44,37,70,34,22,16,43,76,64,7,21,50,82)(3,9,24,51,65,12,6,18,36,67,33,20,41,69,97,32,49,15,39,71,11,26,42,73,95)(10,27,55,86,94,30,25,40,74,96,62,19,46,68,98,29,48,52,83,100,31,54,45,72,101)(28,57,78,102,111,60,56,84,110,112,59,53,77,103,113,61,47,79,99,114,63,81,75,104,116)(58,89,106,119,121,90,88,108,117,122,91,87,105,120,123,92,85,107,118,124,93,80,109,115,125)(126,127,128,129,130,131,132);(1,3,10,28,58)(2,6,19,47,80)(4,12,30,60,90)(5,15,40,75,105)(7,9,25,53,85)(8,20,48,81,89)(11,31,63,93,64)(13,33,62,59,91)(14,36,68,99,115)(16,18,45,77,106)(17,42,46,78,107)(21,41,55,79,108)(22,26,27,56,87)(23,49,54,57,88)(24,52,84,109,44)(29,61,92,34,32)(35,65,94,111,121)(37,39,72,102,117)(38,69,83,104,119)(43,73,86,110,120)(50,51,74,103,118)(66,67,96,112,122)(70,97,98,113,123)(71,100,114,124,76)(82,95,101,116,125);(3,11,32,33,12)(6,9,26,49,20)(10,29,30,31,62)(15,41,18,24,42)(19,27,48,25,54)(28,59,63,60,61)(36,51,73,39,69)(40,45,46,55,52)(47,57,53,81,56)(58,90,91,92,93)(65,95,71,97,67)(68,86,83,74,72)(75,84,79,78,77)(80,89,88,87,85)(94,100,96,101,98)(99,102,103,104,110)(105,107,109,106,108)(111,113,116,112,114)(115,119,117,120,118)(121,122,123,124,125)1043 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21875, 67);