D₁₆₂₀₅ #44569393

order	32410
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₆₂₀₅
unambiguous semidirect product	C₅⋊D₃₂₄₁
unambiguous semidirect product	C₇⋊D₂₃₁₅
unambiguous semidirect product	C₄₆₃⋊D₃₅

Splittings

Fitting, socle, derived, Hall	C₅×C₇×C₄₆₃ ⋊ C₂
Hall	C₇×C₄₆₃ ⋊ D₅
Hall	C₅×C₄₆₃ ⋊ D₇
Sylow	C₄₆₃ ⋊ D₃₅
Hall	C₅×C₇ ⋊ D₄₆₃
Sylow	C₇ ⋊ D₂₃₁₅
Sylow	C₅ ⋊ D₃₂₄₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₇×C₄₆₃	C₂
LCS	G ▷ C₅×C₇×C₄₆₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 475
transitive degree	μ_t(G) = 16205

A minimal faithful permutation representation — 2 generators on 475 points:

(1,2,3,4,5)(6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475);(2,5)(3,4)(7,12)(8,11)(9,10)(14,475)(15,474)(16,473)(17,472)(18,471)(19,470)(20,469)(21,468)(22,467)(23,466)(24,465)(25,464)(26,463)(27,462)(28,461)(29,460)(30,459)(31,458)(32,457)(33,456)(34,455)(35,454)(36,453)(37,452)(38,451)(39,450)(40,449)(41,448)(42,447)(43,446)(44,445)(45,444)(46,443)(47,442)(48,441)(49,440)(50,439)(51,438)(52,437)(53,436)(54,435)(55,434)(56,433)(57,432)(58,431)(59,430)(60,429)(61,428)(62,427)(63,426)(64,425)(65,424)(66,423)(67,422)(68,421)(69,420)(70,419)(71,418)(72,417)(73,416)(74,415)(75,414)(76,413)(77,412)(78,411)(79,410)(80,409)(81,408)(82,407)(83,406)(84,405)(85,404)(86,403)(87,402)(88,401)(89,400)(90,399)(91,398)(92,397)(93,396)(94,395)(95,394)(96,393)(97,392)(98,391)(99,390)(100,389)(101,388)(102,387)(103,386)(104,385)(105,384)(106,383)(107,382)(108,381)(109,380)(110,379)(111,378)(112,377)(113,376)(114,375)(115,374)(116,373)(117,372)(118,371)(119,370)(120,369)(121,368)(122,367)(123,366)(124,365)(125,364)(126,363)(127,362)(128,361)(129,360)(130,359)(131,358)(132,357)(133,356)(134,355)(135,354)(136,353)(137,352)(138,351)(139,350)(140,349)(141,348)(142,347)(143,346)(144,345)(145,344)(146,343)(147,342)(148,341)(149,340)(150,339)(151,338)(152,337)(153,336)(154,335)(155,334)(156,333)(157,332)(158,331)(159,330)(160,329)(161,328)(162,327)(163,326)(164,325)(165,324)(166,323)(167,322)(168,321)(169,320)(170,319)(171,318)(172,317)(173,316)(174,315)(175,314)(176,313)(177,312)(178,311)(179,310)(180,309)(181,308)(182,307)(183,306)(184,305)(185,304)(186,303)(187,302)(188,301)(189,300)(190,299)(191,298)(192,297)(193,296)(194,295)(195,294)(196,293)(197,292)(198,291)(199,290)(200,289)(201,288)(202,287)(203,286)(204,285)(205,284)(206,283)(207,282)(208,281)(209,280)(210,279)(211,278)(212,277)(213,276)(214,275)(215,274)(216,273)(217,272)(218,271)(219,270)(220,269)(221,268)(222,267)(223,266)(224,265)(225,264)(226,263)(227,262)(228,261)(229,260)(230,259)(231,258)(232,257)(233,256)(234,255)(235,254)(236,253)(237,252)(238,251)(239,250)(240,249)(241,248)(242,247)(243,246)(244,245)

Polycyclic presentation:

2,5,7,463|b^a=b^4,c^a=c^6,d^a=d^462

Conjugacy classes & character table

8104 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₆₂₀₅
maximal normal subgroup of	C₂×D₁₆₂₀₅
maximal quotient of	Dic₁₆₂₀₅, C₂×D₁₆₂₀₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32410, 9);

D16205 #44569393