Finite Group DatabasebetaFinite Group Databasebeta| order | 13220 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | Dic661⋊C5 |
| derived, Sylow | C661 ⋊ C4×C5 |
| Hall | C661⋊C5 ⋊ C4 |
| Hall | Dic661 ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D661⋊C5 |
| commutator, hypocenter | C661 | C4×C5 |
| Fitting, socle, cosocle | C2×C661 | C2×C5 |
| prime radical | C2×C661⋊C5 | C2 |
| LCS | G ▷ C661 | |
| UCS | C1 ◁ C2 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 665 |
| transitive degree | μt(G) = 2644 |
A minimal faithful permutation representation — 2 generators on 665 points:
(2,6,7,25,30,3,4,12,14,54)(5,19,20,87,92,9,10,41,43,172)(8,35,36,37,65,16,17,75,77,79)(11,48,49,206,211,22,23,103,105,390)(13,59,60,250,233,27,28,126,128,116)(15,69,70,66,67,32,33,71,31,78)(18,81,82,261,183,38,39,159,161,349)(21,97,98,366,139,45,46,193,195,277)(24,110,111,420,424,51,52,222,224,602)(26,120,121,55,235,56,57,240,117,118)(29,133,134,466,470,62,63,265,267,327)(34,145,146,184,185,72,73,283,93,351)(40,166,167,527,399,84,85,317,319,586)(42,177,178,541,437,89,90,337,339,234)(44,187,188,140,141,94,95,356,68,278)(47,200,201,385,350,100,101,379,381,553)(50,216,217,203,204,107,108,409,102,386)(53,229,230,454,544,113,114,320,340,591)(58,245,246,173,526,123,124,447,328,329)(61,259,260,80,304,130,131,352,156,157)(64,272,273,231,450,136,137,481,433,434)(74,289,290,400,401,148,149,495,212,587)(76,297,99,374,375,152,153,197,198,236)(83,314,242,243,151,163,164,122,370,295)(86,323,324,426,605,169,170,225,529,537)(88,331,332,119,440,174,175,419,237,238)(91,344,274,275,322,180,181,138,484,485)(96,362,363,353,354,190,191,567,186,557)(104,394,395,654,249,208,209,583,572,438)(106,403,404,171,279,213,214,589,142,326)(109,415,416,276,554,219,220,598,486,487)(112,428,389,642,387,226,227,576,509,483)(115,282,312,256,257,144,232,162,129,358)(125,422,423,391,446,248,223,603,578,579)(127,285,359,360,302,252,253,189,255,155)(132,461,228,460,608,262,263,430,431,623)(135,475,476,571,517,269,270,628,612,380)(143,286,287,371,372,280,147,251,196,241)(150,501,455,489,338,292,293,620,425,535)(154,510,218,595,316,299,300,412,413,525)(158,518,341,342,588,305,306,179,528,545)(160,511,477,600,611,309,310,417,631,439)(165,521,508,202,556,315,311,298,383,384)(168,397,398,388,296,321,210,333,205,506)(176,396,538,330,307,334,335,585,534,519)(182,367,551,318,247,346,347,514,524,449)(192,563,451,550,448,357,365,617,402,615)(194,345,348,573,592,368,369,549,408,441)(199,564,488,494,558,376,377,288,459,382)(207,581,580,239,539,392,393,655,442,336)(215,540,410,559,560,406,407,496,355,480)(221,474,291,467,411,418,281,498,499,593)(244,512,513,462,601,445,301,308,429,435)(254,619,503,504,626,452,453,294,471,555)(258,570,505,594,609,456,457,490,536,432)(264,427,532,625,614,463,464,607,469,561)(266,497,548,421,515,468,443,405,543,303)(268,502,627,577,610,472,473,562,436,644)(271,414,632,575,660,478,479,596,651,622)(284,635,636,649,613,491,492,542,630,444)(313,493,569,565,566,522,458,364,361,373)(325,646,647,656,507,530,531,582,643,641)(343,604,590,624,520,546,547,523,657,618)(378,653,650,659,616,568,574,637,645,661)(465,633,634,606,648,516,482,621,533,584)(500,629,599,552,652,638,639,658,640,597)(662,663,664,665);(1,2,5,11,24,53,115,70,69,143,281,472,265,467,363,569,362,568,376,197,97,194,306,238,441,612,195,370,521,309,39,83,165,316,526,608,660,494,131,261,460,624,453,492,637,636,178,340,452,252,28,61,132,264,465,515,302,67,75,151,296,507,634,620,360,250,374,573,406,213,103,207,293,502,632,661,615,447,397,208,23,50,109,221,419,601,433,229,124,247,450,211,399,217,411,594,175,333,537,602,181,345,550,400,206,391,580,610,310,510,326,533,529,538,354,558,379,272,482,531,423,604,649,628,631,339,543,111,225,427,275,384,260,351,555,607,530,323,513,82,162,313,523,214,405,546,341,177,338,534,328,43,93,186,355,561,579,445,242,120,232,436,576,388,203,100,45,19,42,91,182,348,552,638,498,289,496,108,218,414,597,377,257,356,562,273,483,386,277,79,54,116,234,438,248,123,56,7,15,34,74,150,294,505,146,285,493,456,256,126,251,451,618,519,305,156,77,35,76,154,301,514,611,375,235,304,517,101,202,385,575,365,570,653,639,270,477,415,491,73,147,288,486,274,136,62,27,4,8,18,40,86,171,140,66,25,55,119,239,443,495,565,359,187,357,190,94,41,88,176,336,540,613,372,240,314,524,85,168,322,470,92,183,350,554,263,462,428,404,591,416,599,659,342,527,625,468,63,135,271,480,395,224,425,141,153,298,509,643,647,581,566,371,193,367,572,648,209,396,504,466,595,545,587,230,317,422,52,112,228,432,609,430,226,110,223,167,320,401,588,413,267,471,335,394,584,654,347,98,196,373,393,582,656,642,508,297,278,489,420,583,560,478,269,133,266,469,319,528,645,658,598,544,589,227,429,461,487,553,349,172,327,485,321,166,318,164,121,241,444,407,539,334,174,20,44,96,192,95,189,361,290,497,442,237,117,14,31,68,142,169,84,38,16,6,13,29,64,138,276,488,286,145,284,220,417,475,629,574,457,563,651,381,383,200,380,157,118,236,439,551,512,299,152,17,37,80,158,307,520,617,287,60,129,258,458,253,283,490,503,292,148,72,32,12,26,58,125,249,437,233,30,65,139,204,387,481,627,188,358,564,652,479,412,216,410,149,291,500,640,549,346,180,89,10,21,47,102,205,389,577,282,57,122,244,446,614,559,353,184,87,173,330,535,90,179,343,548,403,590,522,312,159,308,170,325,532,626,185,352,556,484,464,324,222,421,541,600,476,630,547,603,646,633,137,201,382,557,585,426,606,279,300,511,644,655,578,105,212,402,369,344,424,605,398,331,536,593,409,586,390,434,449,245,114,231,435,332,418,219,107,48,104,210,246,448,616,596,473,501,392,49,106,215,408,198,128,255,455,621,641,506,295,36,78,155,303,516,463,262,130,59,127,254,454,337,542,650,635,619,657,431,161,259,459,622,623,329,525,315,163,81,160,311,243,366,571,592,440,518,368,46,99,199,378,191,364,567,499,134,268,474,280,33,71,144,113,51,22,9,3)Polycyclic presentation:
2,5,661,2|a^2=d,c^a=c^660,c^b=c^247152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13220, 2);