Finite Group DatabasebetaFinite Group Databasebeta| order | 27459 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C113 · Schur multiplier (H2) = C1 |
| direct product | C113×#6858 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C113 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C32×C113 |
| prime radical | C33×C113 | C32 |
| LCS | G ▷ C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32×C113 ◁ C33×C113 ◁ G | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 149 |
| transitive degree | μt(G) = 9153 |
A minimal faithful permutation representation — 2 generators on 149 points:
(2,4,6)(10,11,14,13,17,23,21,16,27)(12,18,22,20,15,24,31,26,34)(19,29,32,30,28,33,36,25,35)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149);(1,2,5)(3,6,9)(4,8,7)(10,12,19,13,20,30,21,31,36)(11,15,25,17,26,29,16,18,28)(14,22,32,23,24,33,27,34,35)3955 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27459, 5);