Finite Group DatabasebetaFinite Group Databasebeta| order | 19375 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C52×C31 · Schur multiplier (H2) = C5 |
| direct product | C31×#10597832 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C5×C31 | H(3,𝔽5) |
| commutator, Frattini, cosocle | C52 | C52×C31 |
| prime radical | C31×H(3,𝔽5) | C5 |
| LCS | G ▷ C52 ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C5×C31 ◁ C52×C31 ◁ G | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 156 |
| transitive degree | μt(G) = 3875 |
A minimal faithful permutation representation — 2 generators on 156 points:
(1,2,5,13,31)(3,9,23,47,59)(4,8,16,34,60)(6,17,32,61,11)(7,20,46,76,58)(10,26,50,80,88)(12,22,39,33,64)(14,35,65,57,45)(15,38,70,30,21)(18,42,62,91,56)(19,41,63,90,29)(24,36,68,89,28)(25,37,67,94,87)(27,53,72,96,108)(40,66,95,107,52)(43,73,92,111,106)(44,48,77,93,86)(49,71,97,110,85)(51,78,104,109,55)(54,83,100,116,120)(69,98,113,119,75)(74,103,112,124,125)(79,101,115,123,118)(81,99,117,122,105)(82,102,114,121,84)(126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156);(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)(2,6,18,43,74,8,19,44,75,83,22,45,52,53,82,21,25,26,51,81,7,9,24,49,79)(5,14,36,69,99,16,37,42,72,101,39,23,48,78,103,15,17,40,71,100,20,41,50,73,102)(13,32,62,92,112,34,63,77,98,116,33,35,66,96,114,38,67,80,104,117,46,47,68,97,115)(31,59,88,108,120,60,61,89,109,121,64,90,91,110,122,70,65,93,111,123,76,94,95,113,124)1519 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19375, 9);