D₁₄₇₂₈ #72235892

order	29456
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₄₇₂₈

Splittings

Fitting	C₇×C₈×C₂₆₃ ⋊ C₂
Hall	C₇×C₂₆₃ ⋊ D₈
Sylow	C₂₆₃ ⋊ D₅₆
Sylow	C₇ ⋊ D₂₁₀₄

Subgroups

	subgroup	G/H
center	C₂	D₇₃₆₄
commutator, cosocle	C₄×C₇×C₂₆₃	C₂²
Frattini	C₄	C₂×D₁₈₄₁
Fitting	C₇×C₈×C₂₆₃	C₂
socle	C₂×C₇×C₂₆₃	D₄
hypercenter	C₈	D₁₈₄₁
hypocenter	C₇×C₂₆₃	D₈
LCS	G ▷ C₄×C₇×C₂₆₃ ▷ C₂×C₇×C₂₆₃ ▷ C₇×C₂₆₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 278
transitive degree	μ_t(G) = 14728

A minimal faithful permutation representation — 2 generators on 278 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);(2,8)(3,7)(4,6)(10,15)(11,14)(12,13)(17,278)(18,277)(19,276)(20,275)(21,274)(22,273)(23,272)(24,271)(25,270)(26,269)(27,268)(28,267)(29,266)(30,265)(31,264)(32,263)(33,262)(34,261)(35,260)(36,259)(37,258)(38,257)(39,256)(40,255)(41,254)(42,253)(43,252)(44,251)(45,250)(46,249)(47,248)(48,247)(49,246)(50,245)(51,244)(52,243)(53,242)(54,241)(55,240)(56,239)(57,238)(58,237)(59,236)(60,235)(61,234)(62,233)(63,232)(64,231)(65,230)(66,229)(67,228)(68,227)(69,226)(70,225)(71,224)(72,223)(73,222)(74,221)(75,220)(76,219)(77,218)(78,217)(79,216)(80,215)(81,214)(82,213)(83,212)(84,211)(85,210)(86,209)(87,208)(88,207)(89,206)(90,205)(91,204)(92,203)(93,202)(94,201)(95,200)(96,199)(97,198)(98,197)(99,196)(100,195)(101,194)(102,193)(103,192)(104,191)(105,190)(106,189)(107,188)(108,187)(109,186)(110,185)(111,184)(112,183)(113,182)(114,181)(115,180)(116,179)(117,178)(118,177)(119,176)(120,175)(121,174)(122,173)(123,172)(124,171)(125,170)(126,169)(127,168)(128,167)(129,166)(130,165)(131,164)(132,163)(133,162)(134,161)(135,160)(136,159)(137,158)(138,157)(139,156)(140,155)(141,154)(142,153)(143,152)(144,151)(145,150)(146,149)(147,148)

Polycyclic presentation:

2^4,7,263|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^6,f^a=f^262

Conjugacy classes & character table

7367 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₉₄₅₆
maximal normal subgroup of	D₂₉₄₅₆
maximal quotient of	Dic₁₄₇₂₈, D₂₉₄₅₆

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, 263]);;
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.6^262);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D14728 #72235892