Finite Group DatabasebetaFinite Group Databasebeta| order | 30861 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C34×C127 · Schur multiplier (H2) = C37 |
| direct product | C32×H(3,𝔽3)×C127 |
| unambiguous semidirect product | C32⋊(C33×C127) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C33×C127 | C32 |
| commutator, Frattini, cosocle | C3 | C34×C127 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C33×C127 ◁ G | |
900 conjugacy classes of subgroups.
| minimal degree | μ(G) = 142 |
| transitive degree | μt(G) = 10287 |
A minimal faithful permutation representation — 4 generators on 142 points:
(1,2,3)(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142);(4,5,6);(7,8,10)(9,12,14)(11,15,13);(8,11,12)(10,14,15)12573 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30861, 62);