Dic₁₆₃₆₀ #72301629

order	65440
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₆₃₆₀

Splittings

Hall	C₅×C₄₀₉ ⋊ Q₃₂
Sylow	C₄₀₉ ⋊ Dic₄₀
Sylow	C₅ ⋊ Dic₃₂₇₂

Subgroups

	subgroup	G/H
center	C₂	D₁₆₃₆₀
commutator, cosocle	C₅×C₈×C₄₀₉	C₂²
Frattini	C₈	C₂×D₂₀₄₅
Fitting	C₅×C₁₆×C₄₀₉	C₂
socle, prime radical	C₂×C₅×C₄₀₉	D₈
hypercenter	C₁₆	D₂₀₄₅
hypocenter	C₅×C₄₀₉	Q₃₂
LCS	G ▷ C₅×C₈×C₄₀₉ ▷ C₄×C₅×C₄₀₉ ▷ C₂×C₅×C₄₀₉ ▷ C₅×C₄₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 446
transitive degree	μ_t(G) = 65440

A minimal faithful permutation representation — 2 generators on 446 points:

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

Polycyclic presentation:

2^5,5,409|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^4,g^a=g^408

Conjugacy classes & character table

16363 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 5, 409]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^408);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic16360 #72301629