Finite Group DatabasebetaFinite Group Databasebeta| order | 38637 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C27×C53 · Schur multiplier (H2) = C3 |
| direct product | C53×#10624504 |
| subgroup | G/H | |
|---|---|---|
| center | C27×C53 | H(3,𝔽3) |
| commutator | C32 | C3×C27×C53 |
| Frattini, cosocle | C3×C27 | C32×C53 |
| socle | C3×C53 | C32⋊C27 |
| prime radical | H(3,𝔽3)×C53 | C27 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C27×C53 ◁ C3×C27×C53 ◁ G | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 296 |
| transitive degree | μt(G) = 12879 |
A minimal faithful permutation representation — 2 generators on 296 points:
(1,2,7,4,9,24,17,31,62,5,10,25,18,32,63,52,78,119,20,34,65,54,80,121,108,140,179,6,11,26,19,33,64,53,79,120,21,35,66,55,81,122,109,141,180,57,83,124,111,143,182,171,198,221,22,36,67,56,82,123,110,142,181,58,84,125,112,144,183,172,199,222,113,145,184,173,200,223,217,231,240)(3,12,37,15,40,86,49,92,148,16,41,87,50,93,149,106,157,204,51,94,150,107,158,205,170,210,234,14,39,23,47,90,60,103,154,116,48,91,61,104,155,117,168,208,177,105,156,118,169,209,178,216,236,220,46,8,27,68,29,70,126,75,130,102,30,71,127,76,131,185,138,190,167,77,132,186,139,191,224,197,227)(13,28,59,43,73,114,97,135,174,44,74,115,98,136,175,162,195,218,100,137,176,164,196,219,213,230,239,45,38,69,99,88,128,163,151,187,101,89,129,165,152,188,214,206,225,166,153,189,215,207,226,238,235,241,42,72,85,95,133,146,159,192,201,96,134,147,160,193,202,211,228,232,161,194,203,212,229,233,237,242,243)(244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296);(1,3,13)(2,8,28)(4,15,43)(5,16,44)(6,14,45)(7,23,59)(9,29,73)(10,30,74)(11,12,38)(17,49,97)(18,50,98)(19,47,99)(20,51,100)(21,48,101)(22,46,42)(24,60,114)(25,61,115)(26,27,69)(31,75,135)(32,76,136)(33,40,88)(34,77,137)(35,41,89)(36,39,72)(37,85,67)(52,106,162)(53,103,163)(54,107,164)(55,104,165)(56,68,95)(57,105,166)(58,102,96)(62,116,174)(63,117,175)(64,70,128)(65,118,176)(66,71,129)(78,138,195)(79,92,151)(80,139,196)(81,93,152)(82,90,133)(83,94,153)(84,91,134)(86,146,123)(87,147,125)(108,170,213)(109,168,214)(110,126,159)(111,169,215)(112,127,160)(113,167,161)(119,177,218)(120,130,187)(121,178,219)(122,131,188)(124,132,189)(140,197,230)(141,157,206)(142,154,192)(143,158,207)(144,155,193)(145,156,194)(148,201,181)(149,202,183)(150,203,184)(171,216,238)(172,185,211)(173,186,212)(179,220,239)(180,190,225)(182,191,226)(198,210,235)(199,208,228)(200,209,229)(204,232,222)(205,233,223)(217,224,237)(221,227,241)(231,236,242)(234,243,240)8109 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38637, 64);