C₂×C₃⋊(D₆₀₁⋊C₅) #49115159

order	36060
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₃⋊(D₆₀₁⋊C₅)
unambiguous semidirect product	(C₂×C₃)⋊(D₆₀₁⋊C₅)
unambiguous semidirect product	(C₂×C₆₀₁⋊C₅)⋊D₃
unambiguous semidirect product	(C₂×D₁₈₀₃)⋊C₅

Splittings

Fitting, socle	C₂×C₃×C₆₀₁ ⋊ C₂×C₅
derived, Hall	C₃×C₆₀₁ ⋊ C₂²×C₅
Hall	C₃×C₆₀₁⋊C₅ ⋊ C₂²
Hall	C₂×D₁₈₀₃ ⋊ C₅
Hall	C₆₀₁⋊C₅ ⋊ C₂×D₃
Sylow	C₆₀₁ ⋊ C₂×C₅×D₃
Sylow	C₃ ⋊ C₂×D₆₀₁⋊C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	C₃⋊(D₆₀₁⋊C₅)
commutator, cosocle, hypocenter	C₃×C₆₀₁	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₆₀₁	C₂×C₅
LCS	G ▷ C₃×C₆₀₁
UCS	C₁ ◁ C₂

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 606
transitive degree	μ_t(G) = 3606

A minimal faithful permutation representation — 2 generators on 606 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36060, 36);

C2×C3⋊(D601⋊C5) #49115159