Finite Group DatabasebetaFinite Group Databasebeta| order | 21875 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C54×C7 · Schur multiplier (H2) = C55 |
| direct product | C7×5−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C5×C7 | C54 |
| commutator, Frattini, cosocle | C5 | C54×C7 |
| prime radical | C5×C7×H(3,𝔽5) | C5 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C5×C7 ◁ G | |
2316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 132 |
| transitive degree | μt(G) = 875 |
A minimal faithful permutation representation — 4 generators on 132 points:
(1,2,5,14,35,4,8,17,38,67,13,23,44,37,71,34,22,16,43,76,65,7,21,52,85)(3,9,24,53,86,12,26,42,70,66,33,51,41,36,68,32,20,15,39,72,11,6,18,45,77)(10,27,56,73,100,30,49,40,74,102,63,50,46,78,96,29,19,47,80,97,31,25,54,69,98)(28,58,79,106,113,61,84,81,99,114,60,48,82,101,116,62,55,87,103,111,64,57,75,104,112)(59,91,109,118,122,92,83,110,119,123,93,88,105,120,124,94,89,107,115,125,95,90,108,117,121)(126,127,128,129,130,131,132);(1,3,10,28,59)(2,6,19,48,83)(4,12,30,61,92)(5,15,40,75,105)(7,20,50,84,91)(8,9,25,55,88)(11,31,64,95,65)(13,33,63,60,93)(14,36,69,99,115)(16,42,54,82,109)(17,18,46,79,107)(21,41,56,87,110)(22,51,49,58,90)(23,26,27,57,89)(24,47,81,108,44)(29,62,94,34,32)(35,66,96,111,121)(37,45,74,103,118)(38,39,73,101,117)(43,53,78,104,119)(52,70,80,106,120)(67,68,97,112,122)(71,72,98,113,123)(76,77,100,114,124)(85,86,102,116,125);(3,11,32,33,12)(6,20,51,26,9)(10,29,30,31,63)(15,41,42,24,18)(19,49,25,50,27)(28,60,64,61,62)(36,70,53,45,39)(40,54,46,56,47)(48,57,84,55,58)(59,92,93,94,95)(66,86,77,72,68)(69,78,73,80,74)(75,81,87,79,82)(83,88,89,90,91)(96,100,97,102,98)(99,103,106,101,104)(105,107,108,109,110)(111,113,116,112,114)(115,117,118,119,120)(121,122,123,124,125);(2,7,22,23,8)(5,16,17,21,44)(6,20,51,26,9)(14,37,52,38,43)(15,42,18,41,24)(19,50,49,27,25)(35,67,71,76,85)(36,45,70,39,53)(40,54,46,56,47)(48,84,58,57,55)(66,68,72,77,86)(69,74,80,73,78)(75,82,79,87,81)(83,91,90,89,88)(96,97,98,100,102)(99,103,106,101,104)(105,109,107,110,108)(111,112,113,114,116)(115,118,120,117,119)(121,122,123,124,125)4403 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21875, 76);