D₁₄₀₅₆ #72235650

order	28112
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₄₀₅₆

Splittings

Fitting	C₇×C₈×C₂₅₁ ⋊ C₂
Hall	C₇×C₂₅₁ ⋊ D₈
Sylow	C₂₅₁ ⋊ D₅₆
Sylow	C₇ ⋊ D₂₀₀₈

Subgroups

	subgroup	G/H
center	C₂	D₇₀₂₈
commutator, cosocle	C₄×C₇×C₂₅₁	C₂²
Frattini	C₄	C₂×D₁₇₅₇
Fitting	C₇×C₈×C₂₅₁	C₂
socle	C₂×C₇×C₂₅₁	D₄
hypercenter	C₈	D₁₇₅₇
hypocenter	C₇×C₂₅₁	D₈
LCS	G ▷ C₄×C₇×C₂₅₁ ▷ C₂×C₇×C₂₅₁ ▷ C₇×C₂₅₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 266
transitive degree	μ_t(G) = 14056

A minimal faithful permutation representation — 2 generators on 266 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);(2,8)(3,7)(4,6)(10,15)(11,14)(12,13)(17,266)(18,265)(19,264)(20,263)(21,262)(22,261)(23,260)(24,259)(25,258)(26,257)(27,256)(28,255)(29,254)(30,253)(31,252)(32,251)(33,250)(34,249)(35,248)(36,247)(37,246)(38,245)(39,244)(40,243)(41,242)(42,241)(43,240)(44,239)(45,238)(46,237)(47,236)(48,235)(49,234)(50,233)(51,232)(52,231)(53,230)(54,229)(55,228)(56,227)(57,226)(58,225)(59,224)(60,223)(61,222)(62,221)(63,220)(64,219)(65,218)(66,217)(67,216)(68,215)(69,214)(70,213)(71,212)(72,211)(73,210)(74,209)(75,208)(76,207)(77,206)(78,205)(79,204)(80,203)(81,202)(82,201)(83,200)(84,199)(85,198)(86,197)(87,196)(88,195)(89,194)(90,193)(91,192)(92,191)(93,190)(94,189)(95,188)(96,187)(97,186)(98,185)(99,184)(100,183)(101,182)(102,181)(103,180)(104,179)(105,178)(106,177)(107,176)(108,175)(109,174)(110,173)(111,172)(112,171)(113,170)(114,169)(115,168)(116,167)(117,166)(118,165)(119,164)(120,163)(121,162)(122,161)(123,160)(124,159)(125,158)(126,157)(127,156)(128,155)(129,154)(130,153)(131,152)(132,151)(133,150)(134,149)(135,148)(136,147)(137,146)(138,145)(139,144)(140,143)(141,142)

Polycyclic presentation:

2^4,7,251|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^6,f^a=f^250

Conjugacy classes & character table

7031 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₈₁₁₂
maximal normal subgroup of	D₂₈₁₁₂
maximal quotient of	Dic₁₄₀₅₆, D₂₈₁₁₂

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, 7, 251]);;
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.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^250);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D14056 #72235650