Finite Group DatabasebetaFinite Group Databasebeta| order | 36693 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9×C151 · Schur multiplier (H2) = C3 |
| direct product | C151×#6873 |
| subgroup | G/H | |
|---|---|---|
| center | C9×C151 | H(3,𝔽3) |
| commutator | C32 | C3×C9×C151 |
| Frattini, cosocle | C3×C9 | C32×C151 |
| socle | C3×C151 | C32⋊C9 |
| prime radical | H(3,𝔽3)×C151 | C9 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C9×C151 ◁ C3×C9×C151 ◁ G | |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 232 |
| transitive degree | μt(G) = 12231 |
A minimal faithful permutation representation — 2 generators on 232 points:
(1,2,6,4,9,21,16,27,43,18,8,24,39,26,48,62,51,69,5,10,20,17,28,42,38,52,65)(3,11,29,14,31,54,36,56,73,13,7,19,35,23,41,60,46,64,15,25,47,37,50,68,61,71,79)(12,30,40,33,55,63,58,74,77,34,22,44,59,45,66,76,67,78,32,49,53,57,70,72,75,80,81)(82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232);(1,3,12)(2,7,22)(4,14,33)(5,15,32)(6,19,40)(8,25,49)(9,23,45)(10,11,30)(13,34,18)(16,36,58)(17,37,57)(20,29,53)(21,41,63)(24,47,44)(26,50,70)(27,46,67)(28,31,55)(35,59,39)(38,61,75)(42,54,72)(43,64,77)(48,68,66)(51,71,80)(52,56,74)(60,76,62)(65,73,81)(69,79,78)7701 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36693, 20);