Finite Group DatabasebetaFinite Group Databasebeta| order | 21875 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C53×C7 · Schur multiplier (H2) = C53 |
| direct product | C5×C7×#10597832 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C52×C7 | H(3,𝔽5) |
| commutator, Frattini, cosocle | C52 | C53×C7 |
| prime radical | C5×C7×H(3,𝔽5) | C5 |
| LCS | G ▷ C52 ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C52×C7 ◁ C53×C7 ◁ G | |
312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 4375 |
A minimal faithful permutation representation — 3 generators on 137 points:
(1,2,3,4,5)(131,132,133,134,135,136,137);(6,7,10,18,36)(8,14,28,52,64)(9,13,21,39,65)(11,22,37,66,16)(12,25,51,81,63)(15,31,55,85,93)(17,27,44,38,69)(19,40,70,62,50)(20,43,75,35,26)(23,47,67,96,61)(24,46,68,95,34)(29,41,73,94,33)(30,42,72,99,92)(32,58,77,101,113)(45,71,100,112,57)(48,78,97,116,111)(49,53,82,98,91)(54,76,102,115,90)(56,83,109,114,60)(59,88,105,121,125)(74,103,118,124,80)(79,108,117,129,130)(84,106,120,128,123)(86,104,122,127,110)(87,107,119,126,89);(6,8,15,32,59,9,16,33,60,89,17,34,61,90,110,35,62,91,111,123,63,92,112,124,130)(7,11,23,48,79,13,24,49,80,88,27,50,57,58,87,26,30,31,56,86,12,14,29,54,84)(10,19,41,74,104,21,42,47,77,106,44,28,53,83,108,20,22,45,76,105,25,46,55,78,107)(18,37,67,97,117,39,68,82,103,121,38,40,71,101,119,43,72,85,109,122,51,52,73,102,120)(36,64,93,113,125,65,66,94,114,126,69,95,96,115,127,75,70,98,116,128,81,99,100,118,129)1715 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21875, 63);