Finite Group DatabasebetaFinite Group Databasebeta| order | 33777 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C9×C139 · Schur multiplier (H2) = C32 |
| direct product | C139×#6903 |
| subgroup | G/H | |
|---|---|---|
| center | C27×C139 | C32 |
| commutator | C3 | C32×C9×C139 |
| Frattini, cosocle | C9 | C33×C139 |
| socle | C3×C139 | C32×C9 |
| prime radical | H(3,𝔽3)×C139 | C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C27×C139 ◁ G | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 220 |
| transitive degree | μt(G) = 11259 |
A minimal faithful permutation representation — 3 generators on 220 points:
(1,2,6,4,8,18,14,24,39,5,9,19,15,25,40,33,48,62,16,26,41,34,49,63,55,69,77)(3,7,17,12,22,37,31,46,60,13,23,38,32,47,61,54,68,76,11,21,36,30,45,59,53,67,75)(10,20,35,28,43,57,51,65,73,27,42,56,50,64,72,70,78,80,29,44,58,52,66,74,71,79,81)(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);(1,3,10)(2,7,20)(4,12,28)(5,13,27)(6,17,35)(8,22,43)(9,23,42)(11,29,16)(14,31,51)(15,32,50)(18,37,57)(19,38,56)(21,44,26)(24,46,65)(25,47,64)(30,52,34)(33,54,70)(36,58,41)(39,60,73)(40,61,72)(45,66,49)(48,68,78)(53,71,55)(59,74,63)(62,76,80)(67,79,69)(75,81,77);(3,11,13)(7,21,23)(10,27,29)(12,30,32)(17,36,38)(20,42,44)(22,45,47)(28,50,52)(31,53,54)(35,56,58)(37,59,61)(43,64,66)(46,67,68)(51,70,71)(57,72,74)(60,75,76)(65,78,79)(73,80,81)13761 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33777, 50);