D₅×C₄₀₁⋊C₅ #34095330

order	20050
automorphism	?
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₅×C₄₀₁⋊C₅
unambiguous semidirect product	C₅⋊(C₂×C₄₀₁⋊C₅)
unambiguous semidirect product	(D₅×C₄₀₁)⋊C₅

Splittings

Fitting, socle, derived	C₅×C₄₀₁ ⋊ C₂×C₅
Hall	C₅×C₄₀₁⋊C₅ ⋊ C₂
Sylow	C₄₀₁ ⋊ C₅×D₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₄₀₁	C₂×C₅
LCS	G ▷ C₅×C₄₀₁
UCS	C₁

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 406
transitive degree	μ_t(G) = 2005

A minimal faithful permutation representation — 2 generators on 406 points:

(2,3,5,9,17)(4,7,13,25,48)(6,11,21,40,33)(8,15,29,56,84)(10,19,36,18,34)(12,23,44,81,89)(14,27,52,96,63)(16,31,60,111,189)(20,38,70,49,90)(22,42,77,74,64)(24,46,85,148,147)(26,50,92,35,65)(28,54,100,171,154)(30,58,107,181,116)(32,61,113,108,183)(37,68,41,75,67)(39,72,128,104,151)(43,79,139,143,155)(45,83,145,164,117)(47,87,152,239,285)(51,94,161,91,156)(53,98,167,132,118)(55,102,103,176,233)(57,105,174,66,119)(59,109,185,278,242)(62,114,194,237,322)(69,123,82,95,158)(71,126,97,165,121)(73,130,213,190,286)(76,134,125,93,159)(78,137,136,133,122)(80,141,225,234,238)(86,150,236,273,197)(88,153,241,303,275)(99,169,261,229,243)(101,173,266,254,198)(106,179,271,157,244)(110,187,281,358,321)(112,191,287,120,199)(115,195,291,284,361)(124,204,149,230,268)(127,208,172,264,246)(129,211,182,274,201)(131,214,307,184,276)(135,218,210,178,269)(138,221,223,163,247)(140,224,231,255,202)(142,227,316,325,362)(144,170,250,160,248)(146,180,258,216,203)(162,251,166,256,206)(168,259,220,217,207)(175,188,283,353,294)(177,270,349,209,301)(186,280,200,295,193)(192,289,365,245,329)(196,292,367,394,384)(205,297,240,326,363)(212,304,279,356,331)(215,308,373,324,379)(219,310,306,288,364)(222,311,314,318,350)(226,315,323,354,296)(228,317,377,369,355)(232,277,344,253,332)(235,320,351,249,333)(252,337,265,346,335)(257,339,302,272,352)(260,342,313,263,336)(262,293,319,338,299)(267,347,341,309,300)(282,360,330,382,290)(298,368,328,370,357)(305,371,359,393,391)(312,374,376,380,395)(327,381,396,334,383)(340,386,372,366,397)(343,387,375,378,392)(345,385,348,390,389)(388,400,398,399,401)(403,404)(405,406);(1,2,4,8,16,32,62,115,196,293,36,67,121,201,27,53,99,170,263,345,161,250,335,356,211,303,369,390,75,133,216,114,79,140,181,273,353,382,259,17,33,63,116,11,22,43,80,142,228,92,158,246,331,126,207,299,346,109,186,271,351,38,71,127,209,281,359,165,255,111,190,221,3,6,12,24,47,88,48,89,154,242,68,122,202,296,54,101,174,268,349,391,251,336,185,279,304,370,365,396,134,217,309,195,141,226,274,354,239,325,342,34,64,117,197,23,45,84,147,233,321,159,247,107,182,208,300,287,363,187,282,352,392,72,129,212,305,360,394,256,338,191,288,311,7,14,28,55,103,177,90,155,56,104,123,203,189,285,102,175,269,350,236,323,337,385,280,357,371,367,397,401,218,241,328,292,227,100,172,265,326,380,387,65,118,198,294,46,86,151,238,148,234,248,332,183,275,301,291,364,395,283,152,240,327,130,52,97,166,257,340,339,377,289,366,374,15,30,59,110,188,284,156,243,105,178,204,194,286,362,176,85,149,235,322,25,49,91,157,245,330,316,376,398,310,145,231,319,317,173,267,348,381,399,400,119,113,193,290,87,44,82,144,230,318,333,373,276,355,358,225,314,375,361,81,143,229,214,98,168,260,343,388,386,266,60,112,192,31,21,41,76,135,219,244,307,179,272,297,139,223,313,270,150,237,324,379,50,93,160,249,334,384,171,29,57,106,180,9,18,35,66,120,200,213,306,372,199,77,136,220,153,83,146,232,320,378,383,261,344,389,393,315,13,26,51,95,163,253,308,169,262,278,128,210,302,347,40,74,132,61,42,78,138,222,312,329,167,258,341,368,224,5,10,20,39,73,131,215,94,162,252,70,125,206,264,58,108,184,277,19,37,69,124,205,298,96,164,254,295,137)(402,403,405,406,404)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20050, 12);

D5×C401⋊C5 #34095330