Finite Group DatabasebetaFinite Group Databasebeta| order | 30861 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C127 · Schur multiplier (H2) = C9 |
| direct product | C127×#6896 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C127 | C33 |
| commutator, Frattini, cosocle | C32 | C33×C127 |
| prime radical | C33×C127 | C32 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32×C127 ◁ G | |
120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 145 |
| transitive degree | μt(G) = 10287 |
A minimal faithful permutation representation — 3 generators on 145 points:
(2,6,8)(3,7,9)(11,12,16)(14,17,15)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145);(1,2,7,5,8,9,4,6,3)(10,11,14,13,12,15,18,16,17);(1,3,6,4,9,8,5,7,2)(10,12,17,13,16,14,18,11,15)6477 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30861, 43);