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C₇×#14558883 #30681124

• p-group
• nilpotent
• supersolvable
• metabelian
• solvable
• monomial

Aliases

Subgroups

3772 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 4 generators on 350 points:

(1,2,3,4,5,6,7);(8,9,12,19,33,58,92)(10,14,21,35,60,94,136)(11,15,22,36,61,95,135)(13,23,43,75,116,164,134)(16,26,40,64,98,139,184)(17,27,41,65,99,140,183)(18,28,42,66,100,93,137)(20,37,67,107,154,91,24)(25,45,76,117,165,212,182)(29,49,72,104,144,187,230)(30,50,73,105,145,188,229)(31,51,74,106,146,138,185)(32,52,38,69,59,96,141)(34,62,101,147,57,46,44)(39,68,108,155,202,133,47)(48,78,118,166,213,257,228)(53,82,113,151,192,233,272)(54,83,114,152,193,234,271)(55,84,115,153,194,186,231)(56,85,70,110,97,142,189)(63,102,148,195,90,79,77)(71,109,156,203,248,181,80)(81,120,167,214,258,296,270)(86,124,161,199,238,275,307)(87,125,162,200,239,276,306)(88,126,163,201,240,232,273)(89,127,111,158,143,190,235)(103,149,196,241,132,121,119)(112,157,204,249,289,227,122)(123,169,215,259,297,325,305)(128,173,209,245,280,310,331)(129,174,210,246,281,311,330)(130,175,211,247,282,274,308)(131,176,159,206,191,236,277)(150,197,242,283,180,170,168)(160,205,250,290,321,269,171)(172,217,260,298,326,342,329)(177,221,255,287,315,334,345)(178,222,256,288,316,309,332)(179,223,207,252,237,278,312)(198,243,284,317,226,218,216)(208,251,291,322,340,304,219)(220,262,299,327,343,350,344)(224,265,295,320,337,333,346)(225,266,253,293,279,313,335)(244,285,318,338,268,263,261)(254,292,323,341,349,328,264)(267,302,294,324,314,336,347)(286,319,339,348,303,301,300);(9,13,24,46,52,28,15)(12,20,38,22,23,44,42)(14,25,47,79,85,51,27)(19,34,36,37,66,43,69)(21,39,70,41,45,77,74)(26,48,80,121,127,84,50)(33,59,75,100,67,61,62)(35,63,65,68,106,76,110)(40,71,111,73,78,119,115)(49,81,122,170,176,126,83)(58,93,101,116,95,96,107)(60,97,117,146,108,99,102)(64,103,105,109,153,118,158)(72,112,159,114,120,168,163)(82,123,171,218,223,175,125)(92,135,137,141,147,154,164)(94,138,148,165,140,142,155)(98,143,166,194,156,145,149)(104,150,152,157,201,167,206)(113,160,207,162,169,216,211)(124,172,219,263,266,222,174)(136,183,185,189,195,202,212)(139,186,196,213,188,190,203)(144,191,214,240,204,193,197)(151,198,200,205,247,215,252)(161,208,253,210,217,261,256)(173,220,264,301,302,265,221)(184,229,231,235,241,248,257)(187,232,242,258,234,236,249)(192,237,259,282,250,239,243)(199,244,246,251,288,260,293)(209,254,294,255,262,300,295)(230,271,273,277,283,289,296)(233,274,284,297,276,278,290)(238,279,298,316,291,281,285)(245,286,287,292,320,299,324)(272,306,308,312,317,321,325)(275,309,318,326,311,313,322)(280,314,327,337,323,315,319)(307,330,332,335,338,340,342)(310,333,339,343,334,336,341)(331,345,346,347,348,349,350);(8,10,16,29,53,86,128,11,17,30,54,87,129,177,18,31,55,88,130,178,224,32,56,89,131,179,225,267,57,90,132,180,226,268,303,91,133,181,227,269,304,328,134,182,228,270,305,329,344)(9,14,26,49,82,124,173,15,27,50,83,125,174,221,28,51,84,126,175,222,265,52,85,127,176,223,266,302,46,79,121,170,218,263,301,24,47,80,122,171,219,264,13,25,48,81,123,172,220)(12,21,40,72,113,161,209,22,41,73,114,162,210,255,42,74,115,163,211,256,295,38,70,111,159,207,253,294,44,77,119,168,216,261,300,20,39,71,112,160,208,254,23,45,78,120,169,217,262)(19,35,64,104,151,199,245,36,65,105,152,200,246,287,66,106,153,201,247,288,320,69,110,158,206,252,293,324,34,63,103,150,198,244,286,37,68,109,157,205,251,292,43,76,118,167,215,260,299)(33,60,98,144,192,238,280,61,99,145,193,239,281,315,100,146,194,240,282,316,337,59,97,143,191,237,279,314,62,102,149,197,243,285,319,67,108,156,204,250,291,323,75,117,166,214,259,298,327)(58,94,139,187,233,275,310,95,140,188,234,276,311,334,93,138,186,232,274,309,333,96,142,190,236,278,313,336,101,148,196,242,284,318,339,107,155,203,249,290,322,341,116,165,213,258,297,326,343)(92,136,184,230,272,307,331,135,183,229,271,306,330,345,137,185,231,273,308,332,346,141,189,235,277,312,335,347,147,195,241,283,317,338,348,154,202,248,289,321,340,349,164,212,257,296,325,342,350)

Conjugacy classes & character table

2695 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16807, 81);