Finite Group DatabasebetaFinite Group Databasebeta| order | 34375 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C11×C25 · Schur multiplier (H2) = C5 |
| direct product | C11×#15691082 |
| subgroup | G/H | |
|---|---|---|
| center | C11×C25 | H(3,𝔽5) |
| commutator | C52 | C5×C11×C25 |
| Frattini, cosocle | C5×C25 | C52×C11 |
| socle | C5×C11 | C52⋊C25 |
| prime radical | C11×H(3,𝔽5) | C25 |
| LCS | G ▷ C52 ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C11×C25 ◁ C5×C11×C25 ◁ G | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 636 |
| transitive degree | μt(G) = 6875 |
A minimal faithful permutation representation — 2 generators on 636 points:
(1,2,6,19,52,4,9,22,55,116,16,32,64,122,207,48,84,136,216,314,109,166,234,326,422,18,34,66,54,119,50,86,138,121,213,111,168,236,215,323,201,274,349,325,433,307,386,457,435,523,113,8,28,78,158,203,31,79,159,264,309,83,160,265,378,416,165,266,379,483,508,272,380,484,558,5,10,23,56,115,17,33,65,123,206,49,85,137,217,313,110,167,235,327,421,200,273,348,436,513,51,30,21,61,131,112,82,63,132,228,202,164,135,229,342,308,271,233,343,452,415,385,347,453,537)(3,11,20,57,124,14,36,59,126,219,45,90,129,222,330,106,175,226,335,440,197,282,340,446,528,47,7,24,67,139,108,26,70,141,238,199,76,146,242,351,306,156,249,355,459,414,262,363,463,541,13,41,69,53,114,44,101,145,118,205,105,191,248,211,312,196,299,362,321,420,304,409,470,431,512,15,29,60,120,212,46,81,130,214,322,107,163,227,324,432,198,270,341,434,522,305,384,451,524,582,43,27,62,133,230,104,77,134,231,344,195,157,232,345,454,303,263,346,455,538,413,377,456,539,592)(12,25,71,147,237,39,74,149,251,350,98,154,254,366,458,188,260,369,473,540,296,375,476,552,593,100,73,58,125,204,190,153,128,221,311,298,259,225,334,419,408,374,339,445,511,505,481,450,533,574,42,80,68,140,239,103,162,143,241,352,194,269,246,354,460,302,383,360,462,542,412,487,468,544,594,40,75,144,117,208,99,155,247,210,316,189,261,361,320,425,297,376,469,430,516,407,482,549,521,577,38,35,87,169,218,97,89,171,275,329,187,174,278,387,439,295,281,390,488,527,406,393,491,561,585)(37,91,176,250,328,94,178,284,365,438,183,286,395,472,526,291,397,495,551,584,402,497,566,600,614,93,88,170,220,331,182,173,277,333,442,290,280,389,444,530,401,392,490,532,587,501,493,563,589,616,102,72,148,240,310,193,151,253,353,418,301,256,368,461,510,411,371,475,543,573,507,478,554,595,607,95,152,127,209,317,184,258,224,319,427,292,373,338,429,518,403,480,449,520,579,502,557,536,581,611,96,161,142,243,315,186,268,245,357,424,294,382,359,465,515,405,486,467,546,576,504,560,548,597,609)(92,177,252,332,417,180,285,367,443,509,288,396,474,531,572,399,496,553,588,606,499,567,601,617,621,181,172,223,336,426,289,279,337,447,517,400,391,448,534,578,500,492,535,590,610,569,564,591,618,623,192,150,244,318,423,300,255,358,428,514,410,370,466,519,575,506,477,547,580,608,571,555,598,612,622,179,257,283,356,437,287,372,394,464,525,398,479,494,545,583,498,556,565,596,613,568,602,605,619,624,185,267,276,364,441,293,381,388,471,529,404,485,489,550,586,503,559,562,599,615,570,603,604,620,625)(626,627,628,629,630,631,632,633,634,635,636);(1,3,12,37,92)(2,7,25,72,150)(4,14,39,94,180)(5,15,40,95,179)(6,20,58,127,223)(8,29,80,161,267)(9,26,74,151,255)(10,27,75,91,177)(11,35,88,172,30)(13,42,102,192,113)(16,45,98,183,288)(17,46,99,184,287)(18,47,100,93,181)(19,53,117,209,318)(21,62,71,148,252)(22,59,128,224,337)(23,60,87,170,276)(24,68,142,244,66)(28,69,144,176,283)(31,81,162,268,381)(32,76,154,256,370)(33,77,155,178,285)(34,41,73,152,257)(36,89,173,279,82)(38,96,185,51,43)(44,103,193,300,203)(48,106,188,291,399)(49,107,189,292,398)(50,108,190,182,289)(52,114,204,310,417)(54,120,169,243,356)(55,118,210,319,428)(56,57,125,220,332)(61,67,140,240,336)(63,134,149,253,367)(64,129,225,338,448)(65,130,171,277,388)(70,143,245,358,138)(78,133,147,250,364)(79,145,247,284,394)(83,163,269,382,485)(84,156,260,371,477)(85,157,261,286,396)(86,101,153,258,372)(90,174,280,391,164)(97,186,293,112,104)(105,194,301,410,309)(109,197,296,402,499)(110,198,297,403,498)(111,199,298,290,400)(115,124,218,328,437)(116,205,311,418,509)(119,212,239,317,426)(121,214,275,357,464)(122,211,320,429,519)(123,126,221,333,443)(131,139,237,331,441)(132,141,241,353,447)(135,232,254,368,474)(136,226,339,449,535)(137,227,278,389,489)(146,246,359,466,236)(158,230,208,315,423)(159,231,251,365,471)(160,248,361,395,494)(165,270,383,486,559)(166,262,375,478,555)(167,263,376,397,496)(168,191,259,373,479)(175,281,392,492,271)(187,294,404,202,195)(196,302,411,506,416)(200,305,407,502,568)(201,306,408,401,500)(206,219,329,438,525)(207,312,419,510,572)(213,322,352,427,517)(215,324,387,465,545)(216,321,430,520,580)(217,222,334,444,531)(228,238,350,442,529)(229,242,354,461,534)(233,346,369,475,553)(234,340,450,536,591)(235,341,390,490,562)(249,360,467,547,349)(264,344,316,424,514)(265,345,366,472,550)(266,362,469,495,565)(272,384,487,560,603)(273,377,482,497,567)(274,299,374,480,556)(282,393,493,564,385)(295,405,503,308,303)(304,412,507,571,508)(307,414,505,501,569)(313,330,439,526,583)(314,420,511,573,606)(323,432,460,518,578)(325,434,488,546,596)(326,431,521,581,612)(327,335,445,532,588)(342,351,458,530,586)(343,355,462,543,590)(347,456,476,554,601)(348,451,491,563,604)(363,468,548,598,457)(378,454,425,515,575)(379,455,473,551,599)(380,470,549,566,605)(386,409,481,557,602)(406,504,570,415,413)(421,440,527,584,613)(422,512,574,607,621)(433,522,542,579,610)(435,524,561,597,619)(436,446,533,589,617)(452,459,540,587,615)(453,463,544,595,618)(483,538,516,576,608)(484,539,552,600,620)(513,528,585,614,624)(523,582,594,611,623)(537,541,593,616,625)(558,592,577,609,622)2695 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(34375, 25);