Finite Group DatabasebetaFinite Group Databasebeta| order | 27459 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C113 · Schur multiplier (H2) = C33 |
| direct product | C113×#6908 |
| subgroup | G/H | |
|---|---|---|
| center | C9×C113 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C33×C113 |
| socle | C3×C113 | C3×H(3,𝔽3) |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C9×C113 ◁ C3×C9×C113 ◁ G | |
134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 140 |
| transitive degree | μt(G) = 3051 |
A minimal faithful permutation representation — 3 generators on 140 points:
(1,2,5)(3,7,12)(4,6,13)(8,17,11)(9,15,20)(10,14,21)(16,24,19)(18,22,26)(23,27,25)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140);(2,6,8)(7,14,16)(15,22,23);(1,3,9,4,10,18,11,19,25)(2,7,15,6,14,22,8,16,23)(5,12,20,13,21,26,17,24,27)5763 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27459, 55);