D₃₈₈₈ #72211243

order	7776
automorphism	Aut(G) order 5038848
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₈₈₈

Splittings

Fitting	C₁₆×C₂₄₃ ⋊ C₂
Sylow	C₂₄₃ ⋊ D₁₆

Subgroups

	subgroup	G/H
center	C₂	D₁₉₄₄
commutator, cosocle	C₈×C₂₄₃	C₂²
Frattini	C₈×C₈₁	C₂×D₃
Fitting	C₁₆×C₂₄₃	C₂
socle	C₂×C₃	D₆₄₈
hypercenter	C₁₆	D₂₄₃
hypocenter	C₂₄₃	D₁₆
LCS	G ▷ C₈×C₂₄₃ ▷ C₄×C₂₄₃ ▷ C₂×C₂₄₃ ▷ C₂₄₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 259
transitive degree	μ_t(G) = 3888

A minimal faithful permutation representation — 2 generators on 259 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,259)(19,258)(20,257)(21,256)(22,255)(23,254)(24,253)(25,252)(26,251)(27,250)(28,249)(29,248)(30,247)(31,246)(32,245)(33,244)(34,243)(35,242)(36,241)(37,240)(38,239)(39,238)(40,237)(41,236)(42,235)(43,234)(44,233)(45,232)(46,231)(47,230)(48,229)(49,228)(50,227)(51,226)(52,225)(53,224)(54,223)(55,222)(56,221)(57,220)(58,219)(59,218)(60,217)(61,216)(62,215)(63,214)(64,213)(65,212)(66,211)(67,210)(68,209)(69,208)(70,207)(71,206)(72,205)(73,204)(74,203)(75,202)(76,201)(77,200)(78,199)(79,198)(80,197)(81,196)(82,195)(83,194)(84,193)(85,192)(86,191)(87,190)(88,189)(89,188)(90,187)(91,186)(92,185)(93,184)(94,183)(95,182)(96,181)(97,180)(98,179)(99,178)(100,177)(101,176)(102,175)(103,174)(104,173)(105,172)(106,171)(107,170)(108,169)(109,168)(110,167)(111,166)(112,165)(113,164)(114,163)(115,162)(116,161)(117,160)(118,159)(119,158)(120,157)(121,156)(122,155)(123,154)(124,153)(125,152)(126,151)(127,150)(128,149)(129,148)(130,147)(131,146)(132,145)(133,144)(134,143)(135,142)(136,141)(137,140)(138,139)

Polycyclic presentation:

2^5,3^5|b^2=cd,c^2=de,d^2=e,f^3=g,g^3=h,h^3=i,i^3=j,b^a=bc,c^a=cd,d^a=de,f^a=f^2g^2h^2i^2j^2,g^a=g^2h^2i^2j^2,h^a=h^2i^2j^2,i^a=i^2j^2,j^a=j^2

Conjugacy classes & character table

1947 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₇₇₇₆, D₁₁₆₆₄, D₁₉₄₄₀, …
maximal normal subgroup of	D₇₇₇₆
maximal quotient of	D₇₇₇₆, Dic₃₈₈₈, D₁₁₆₆₄, …

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", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 6, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
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*F.7^2*F.8^2*F.9^2*F.10^2);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2*F.9^2*F.10^2);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.9^2*F.10^2);;
gap> SetConjugate(coll, 9, 1, F.9^2*F.10^2);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D3888 #72211243