D₁₄₁₂₀ #72235668

order	28240
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₄₁₂₀

Splittings

Fitting	C₅×C₈×C₃₅₃ ⋊ C₂
Hall	C₅×C₃₅₃ ⋊ D₈
Sylow	C₃₅₃ ⋊ D₄₀
Sylow	C₅ ⋊ D₂₈₂₄

Subgroups

	subgroup	G/H
center	C₂	D₇₀₆₀
commutator, cosocle	C₄×C₅×C₃₅₃	C₂²
Frattini	C₄	C₂×D₁₇₆₅
Fitting	C₅×C₈×C₃₅₃	C₂
socle	C₂×C₅×C₃₅₃	D₄
hypercenter	C₈	D₁₇₆₅
hypocenter	C₅×C₃₅₃	D₈
LCS	G ▷ C₄×C₅×C₃₅₃ ▷ C₂×C₅×C₃₅₃ ▷ C₅×C₃₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 366
transitive degree	μ_t(G) = 14120

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

Polycyclic presentation:

2^4,5,353|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^4,f^a=f^352

Conjugacy classes & character table

7063 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₈₂₄₀
maximal normal subgroup of	D₂₈₂₄₀
maximal quotient of	Dic₁₄₁₂₀, D₂₈₂₄₀

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 5, 353]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^352);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D14120 #72235668