Finite Group DatabasebetaFinite Group Databasebeta| order | 30861 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C127 · Schur multiplier (H2) = C3 |
| direct product | C127×#6882 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C127 | #732 |
| commutator, Frattini, cosocle | C3×C9 | C32×C127 |
| prime radical | C32×C127 | H(3,𝔽3) |
| LCS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C127 ◁ C32×C127 ◁ C3×C9×C127 ◁ G | |
38 conjugacy classes of subgroups.
| minimal degree | μ(G) = 154 |
| transitive degree | μt(G) = 3429 |
A minimal faithful permutation representation — 2 generators on 154 points:
(1,2,5,4,9,15,13,8,21)(3,10,14,11,7,19,22,20,27)(6,16,23,18,25,24,17,26,12)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154);(2,6,7,9,18,20,8,17,10)(3,11,22)(5,14,25,21,27,16,15,19,26)(12,24,23)2413 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30861, 29);