Finite Group DatabasebetaFinite Group Databasebeta| order | 30861 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C27×C127 · Schur multiplier (H2) = C32 |
| direct product | C127×C32⋊C27 |
| unambiguous semidirect product | C32⋊(C27×C127) |
| subgroup | G/H | |
|---|---|---|
| center | C3×C9×C127 | C32 |
| commutator | C3 | C3×C27×C127 |
| Frattini, cosocle | C3×C9 | C32×C127 |
| socle | C32×C127 | C3×C9 |
| prime radical | C33×C127 | C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C9×C127 ◁ G | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 10287 |
A minimal faithful permutation representation — 2 generators on 163 points:
(1,2,4)(3,6,8)(5,9,7)(10,11,14,12,15,20,17,22,27,13,16,21,18,23,28,25,30,33,19,24,29,26,31,34,32,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163);(2,5,6)(4,8,9)12573 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30861, 12);