Finite Group DatabasebetaFinite Group Databasebeta| order | 21875 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C52×C7 · Schur multiplier (H2) = C5 |
| direct product | C7×#15691089 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C5×C7 | H(3,𝔽5)⋊C5 |
| commutator, Frattini, cosocle | C53 | C52×C7 |
| prime radical | C7×H(3,𝔽5)⋊C5 | C5 |
| LCS | G ▷ C53 ▷ C52 ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C5×C7 ◁ C52×C7 ◁ C53×C7 ◁ G | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 132 |
| transitive degree | μt(G) = 875 |
A minimal faithful permutation representation — 2 generators on 132 points:
(1,2,5,13,31)(3,9,23,32,59)(4,7,16,33,60)(6,17,40,70,87)(8,21,47,78,58)(10,26,41,61,88)(11,25,14,35,64)(12,20,39,34,63)(15,38,69,30,22)(18,42,73,101,86)(19,45,66,93,57)(24,49,71,98,107)(27,53,79,89,108)(28,52,48,65,91)(29,46,37,62,90)(36,67,94,56,44)(43,75,95,112,85)(50,68,96,114,106)(51,74,99,117,119)(54,83,102,109,120)(55,77,72,92,110)(76,104,111,121,84)(80,100,113,122,105)(81,103,115,123,118)(82,97,116,124,125)(126,127,128,129,130,131,132);(1,3,10,27,54,4,11,28,55,84,12,29,56,85,105,30,57,86,106,118,58,87,107,119,125)(2,6,18,43,76,7,9,24,50,80,20,25,26,51,81,22,46,52,53,82,8,19,44,77,83)(5,14,36,68,97,16,37,42,74,102,39,45,49,79,104,15,17,41,72,100,21,23,48,75,103)(13,32,61,89,109,33,35,65,92,111,34,62,67,95,113,38,66,73,96,115,47,40,71,99,116)(31,59,88,108,120,60,64,91,110,121,63,90,94,112,122,69,93,101,114,123,78,70,98,117,124)1043 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21875, 32);