D₁₄₀₆₄ #72235652

order	28128
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₄₀₆₄

Splittings

Fitting	C₃×C₁₆×C₂₉₃ ⋊ C₂
Hall	C₃×C₂₉₃ ⋊ D₁₆
Sylow	C₂₉₃ ⋊ D₄₈
Sylow	C₃ ⋊ D₄₆₈₈

Subgroups

	subgroup	G/H
center	C₂	D₇₀₃₂
commutator, cosocle	C₃×C₈×C₂₉₃	C₂²
Frattini	C₈	C₂×D₈₇₉
Fitting	C₃×C₁₆×C₂₉₃	C₂
socle	C₂×C₃×C₂₉₃	D₈
hypercenter	C₁₆	D₈₇₉
hypocenter	C₃×C₂₉₃	D₁₆
LCS	G ▷ C₃×C₈×C₂₉₃ ▷ C₃×C₄×C₂₉₃ ▷ C₂×C₃×C₂₉₃ ▷ C₃×C₂₉₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 312
transitive degree	μ_t(G) = 14064

A minimal faithful permutation representation — 2 generators on 312 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,19)(21,312)(22,311)(23,310)(24,309)(25,308)(26,307)(27,306)(28,305)(29,304)(30,303)(31,302)(32,301)(33,300)(34,299)(35,298)(36,297)(37,296)(38,295)(39,294)(40,293)(41,292)(42,291)(43,290)(44,289)(45,288)(46,287)(47,286)(48,285)(49,284)(50,283)(51,282)(52,281)(53,280)(54,279)(55,278)(56,277)(57,276)(58,275)(59,274)(60,273)(61,272)(62,271)(63,270)(64,269)(65,268)(66,267)(67,266)(68,265)(69,264)(70,263)(71,262)(72,261)(73,260)(74,259)(75,258)(76,257)(77,256)(78,255)(79,254)(80,253)(81,252)(82,251)(83,250)(84,249)(85,248)(86,247)(87,246)(88,245)(89,244)(90,243)(91,242)(92,241)(93,240)(94,239)(95,238)(96,237)(97,236)(98,235)(99,234)(100,233)(101,232)(102,231)(103,230)(104,229)(105,228)(106,227)(107,226)(108,225)(109,224)(110,223)(111,222)(112,221)(113,220)(114,219)(115,218)(116,217)(117,216)(118,215)(119,214)(120,213)(121,212)(122,211)(123,210)(124,209)(125,208)(126,207)(127,206)(128,205)(129,204)(130,203)(131,202)(132,201)(133,200)(134,199)(135,198)(136,197)(137,196)(138,195)(139,194)(140,193)(141,192)(142,191)(143,190)(144,189)(145,188)(146,187)(147,186)(148,185)(149,184)(150,183)(151,182)(152,181)(153,180)(154,179)(155,178)(156,177)(157,176)(158,175)(159,174)(160,173)(161,172)(162,171)(163,170)(164,169)(165,168)(166,167)

Polycyclic presentation:

2^5,3,293|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^2,g^a=g^292

Conjugacy classes & character table

7035 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₈₁₂₈
maximal normal subgroup of	D₂₈₁₂₈
maximal quotient of	Dic₁₄₀₆₄, D₂₈₁₂₈

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, 3, 293]);;
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.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^292);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D14064 #72235652