Finite Group DatabasebetaFinite Group Databasebeta| order | 33777 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9×C139 · Schur multiplier (H2) = C3 |
| direct product | C139×#6869 |
| subgroup | G/H | |
|---|---|---|
| center | C9×C139 | H(3,𝔽3) |
| commutator | C32 | C3×C9×C139 |
| Frattini, cosocle | C3×C9 | C32×C139 |
| socle | C3×C139 | C32⋊C9 |
| prime radical | C33×C139 | C9 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C9×C139 ◁ C3×C9×C139 ◁ G | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 166 |
| transitive degree | μt(G) = 3753 |
A minimal faithful permutation representation — 2 generators on 166 points:
(1,2,5,3,8,13,9,17,21,4,6,12,10,14,20,18,22,26,11,7,15,19,16,23,25,24,27)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166);(2,6,7)(8,14,16)(17,22,24)7089 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33777, 16);