Finite Group DatabasebetaFinite Group Databasebeta| order | 21875 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C53×C7 · Schur multiplier (H2) = C53 |
| direct product | C7×#15691126 |
| 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)(3,9,24,51,65)(4,8,17,38,66)(6,18,36,67,12)(7,21,50,82,64)(10,27,55,86,94)(11,26,42,73,95)(13,23,44,37,70)(15,39,71,32,49)(16,43,76,34,22)(19,46,68,98,62)(20,41,69,97,33)(25,40,74,96,30)(28,57,78,102,111)(29,48,52,83,100)(31,54,45,72,101)(47,79,99,114,61)(53,77,103,113,59)(56,84,110,112,60)(58,89,106,119,121)(63,81,75,104,116)(80,109,115,125,93)(85,107,118,124,92)(87,105,120,123,91)(88,108,117,122,90)(126,127,128,129,130,131,132);(1,3,10,28,58,13,33,62,59,91,64,11,31,63,93,4,12,30,60,90,34,32,29,61,92)(2,6,19,47,80,23,49,54,57,88,7,9,25,53,85,8,20,48,81,89,22,26,27,56,87)(5,15,40,75,105,44,24,52,84,109,21,41,55,79,108,17,42,46,78,107,16,18,45,77,106)(14,36,68,99,115,37,39,72,102,117,50,51,74,103,118,38,69,83,104,119,43,73,86,110,120)(35,65,94,111,121,70,97,98,113,123,82,95,101,116,125,66,67,96,112,122,76,71,100,114,124);(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, 69);