Finite Group DatabasebetaFinite Group Databasebeta| order | 36207 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C149 · Schur multiplier (H2) = C33 |
| direct product | C149×C9⋊H(3,𝔽3) |
| unambiguous semidirect product | C9⋊(H(3,𝔽3)×C149) |
| unambiguous semidirect product | (C9×C149)⋊H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C149 | C33 |
| commutator, Frattini, cosocle | C32 | C33×C149 |
| prime radical | C3×H(3,𝔽3)×C149 | C3 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32×C149 ◁ G | |
166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 12069 |
A minimal faithful permutation representation — 3 generators on 167 points:
(2,4,7)(5,9,8)(10,11,13)(12,15,17)(14,18,16)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167);(1,2,5,3,4,8,6,7,9);(11,14,15)(13,17,18)7599 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36207, 39);