#30681057

order	16807
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₇² · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₇²	H(3,𝔽₇)
commutator, Frattini, prime radical, cosocle	C₇³	C₇²
LCS	G ▷ C₇³ ▷ C₇² ▷ C₁
UCS	C₁ ◁ C₇² ◁ C₇³ ◁ G
LECS	G ▷ C₇³ ▷ C₇² ▷ C₁

45 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 686
transitive degree	μ_t(G) = 2401

A minimal faithful permutation representation — 2 generators on 686 points:

(1,2,5,14,37,77,119,4,8,17,40,80,122,178,13,24,47,86,128,121,180,36,59,46,95,79,126,185,76,58,55,39,84,135,195,118,23,16,44,94,149,206,176,7,21,53,103,160,217)(3,9,25,60,78,123,182,12,22,56,89,125,181,234,35,6,18,38,81,129,187,75,20,45,83,127,186,238,74,52,54,104,131,190,244,34,27,15,41,87,138,196,11,32,43,85,136,120,177)(10,28,61,82,132,192,239,31,64,92,134,191,245,281,71,50,48,96,139,197,249,70,26,62,88,140,179,232,73,57,105,137,124,183,231,30,19,42,90,143,201,235,33,51,93,142,130,188,240)(29,63,91,145,189,242,273,68,106,147,133,193,241,272,114,49,98,153,210,246,276,116,101,148,150,198,251,283,67,65,97,141,199,250,282,69,100,107,162,184,233,274,72,102,152,144,202,236,277)(66,99,155,207,259,285,313,112,157,151,200,252,284,304,172,109,163,219,253,287,307,111,158,209,203,237,279,303,113,159,146,204,255,290,302,115,108,164,194,243,275,305,117,161,154,211,247,278,308)(110,167,208,260,286,315,326,170,166,220,254,288,310,329,169,215,262,269,298,319,333,171,216,205,256,291,306,325,173,165,221,257,292,309,328,174,218,212,248,280,311,324,175,156,213,263,294,314,323)(168,224,265,296,317,330,337,225,223,261,289,318,332,340,226,266,270,300,320,334,342,227,267,297,299,321,335,343,228,222,258,293,312,331,339,229,268,271,301,322,336,341,230,214,264,295,316,327,338)(344,345,349,360,384,421,466,347,351,362,386,423,468,518,348,352,363,387,424,469,519,358,368,391,428,472,522,574,359,369,392,429,473,523,575,382,402,435,479,528,579,616,383,403,436,480,529,580,617)(346,353,370,404,450,501,559,356,372,390,432,483,467,520,357,373,406,385,425,474,530,381,350,364,393,437,490,548,355,366,395,427,476,532,517,379,367,396,439,422,470,524,380,401,361,388,430,481,539)(354,374,410,438,491,525,582,376,409,394,440,471,526,584,377,365,397,443,482,540,576,417,399,442,431,484,521,577,378,400,389,433,486,531,588,419,449,405,451,475,533,573,420,371,407,426,477,535,581)(375,412,457,493,536,589,618,415,408,434,488,545,590,619,416,411,452,485,527,586,626,463,455,453,487,541,583,615,418,398,445,496,542,585,622,414,447,441,478,537,593,623,464,448,444,492,534,578,620)(413,456,497,549,600,624,651,461,505,494,550,587,625,648,460,446,495,546,596,633,646,513,500,507,543,597,621,647,465,499,489,544,594,629,654,462,458,502,553,591,630,645,515,454,503,538,592,627,650)(459,504,560,599,637,661,669,511,506,561,607,634,649,667,512,556,554,595,631,657,668,570,498,551,603,632,658,675,516,557,552,604,640,655,666,514,558,564,601,628,652,671,510,508,547,598,636,653,672)(509,555,602,635,656,670,680,567,609,612,642,663,677,684,568,610,613,643,664,678,685,566,565,608,641,662,676,683,572,562,605,638,659,673,681,571,563,606,639,660,674,682,569,611,614,644,665,679,686);(1,3,10,29,66,110,168)(2,6,19,49,99,156,214)(4,12,31,68,112,170,225)(5,15,42,91,146,205,258)(7,22,57,106,161,218,268)(8,20,51,101,157,167,224)(9,26,63,108,165,222,23)(11,33,72,117,175,230,176)(13,35,71,114,172,169,226)(14,38,82,133,194,248,289)(16,45,62,107,163,220,270)(17,43,93,147,164,221,271)(18,48,97,151,208,261,55)(21,54,105,152,209,262,297)(24,52,28,65,109,166,223)(25,61,98,154,212,264,47)(27,64,100,158,215,266,59)(30,69,115,174,229,118,34)(32,50,102,159,216,267,58)(36,75,70,116,111,171,227)(37,78,124,184,237,280,312)(39,85,137,162,219,269,301)(40,83,134,153,211,263,300)(41,88,141,200,254,293,95)(44,60,90,144,203,256,295)(46,56,92,148,155,213,265)(53,89,142,145,204,257,296)(67,113,173,228,76,74,73)(77,120,179,233,275,306,327)(79,127,132,193,247,288,317)(80,125,143,202,255,294,322)(81,130,189,243,286,316,128)(84,131,191,210,259,298,318)(86,104,96,150,207,260,299)(87,139,198,252,291,320,94)(103,136,140,199,253,292,321)(119,177,231,272,302,323,337)(121,181,201,242,285,311,332)(122,123,183,236,278,309,330)(126,129,188,241,284,314,334)(135,186,192,246,287,315,335)(138,197,250,290,319,336,160)(149,190,245,251,279,310,331)(178,182,235,276,305,326,340)(180,234,240,283,308,329,342)(185,187,239,282,313,333,343)(195,238,281,274,304,325,339)(196,232,273,303,324,338,217)(206,244,249,277,307,328,341)(344,346,354,375,413,459,509,348,357,377,416,460,512,568,359,355,378,418,465,516,572,383,380,420,464,515,510,569,347,356,376,415,461,511,567,358,381,417,463,513,570,566,382,379,419,414,462,514,571)(345,350,365,398,446,498,555,352,367,400,448,499,558,610,369,353,371,408,454,504,562,403,373,409,455,505,556,611,351,366,399,447,500,557,609,368,401,449,412,458,508,565,402,372,374,411,456,506,563)(349,361,389,434,489,547,602,363,390,407,453,503,561,613,392,364,394,441,494,551,605,436,396,442,457,507,564,614,362,370,405,452,502,560,612,391,406,410,445,497,554,608,435,395,397,444,495,552,606)(360,385,426,478,538,595,635,387,427,440,493,550,604,643,429,388,431,485,543,598,638,480,432,451,496,553,607,644,386,393,438,492,549,603,642,428,439,443,488,546,601,641,479,404,433,487,544,599,639)(384,422,471,527,587,628,656,424,450,484,542,597,637,664,473,425,475,534,591,631,659,529,476,491,545,600,640,665,423,430,482,541,596,636,663,472,483,486,537,594,634,662,528,437,477,536,592,632,660)(421,467,521,578,621,649,670,469,490,533,590,630,658,678,523,470,525,583,624,652,673,580,501,540,593,633,661,679,468,474,531,589,629,657,677,522,532,535,586,627,655,676,579,481,526,585,625,653,674)(466,517,573,615,645,666,680,519,539,582,623,651,672,685,575,520,576,618,646,667,681,617,548,588,626,654,675,686,518,524,581,622,650,671,684,574,559,584,620,648,669,683,616,530,577,619,647,668,682)

Polycyclic presentation:

7^5|a^7=d,b^7=e,b^a=bc,c^a=cde^3,c^b=cde

Conjugacy classes & character table

391 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16807, 14);