D₇₇₄₄ #72212907

order	15488
automorphism	Aut(G) order 27258880
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₇₇₄₄

Splittings

Fitting	C₆₄×C₁₂₁ ⋊ C₂
Sylow	C₁₂₁ ⋊ D₆₄

Subgroups

	subgroup	G/H
center	C₂	D₃₈₇₂
commutator, cosocle	C₃₂×C₁₂₁	C₂²
Frattini	C₁₁×C₃₂	C₂×D₁₁
Fitting	C₆₄×C₁₂₁	C₂
socle	C₂×C₁₁	D₃₅₂
hypercenter	C₆₄	D₁₂₁
hypocenter	C₁₂₁	D₆₄
LCS	G ▷ C₃₂×C₁₂₁ ▷ C₁₆×C₁₂₁ ▷ C₈×C₁₂₁ ▷ C₄×C₁₂₁ ▷ C₂×C₁₂₁ ▷ C₁₂₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ C₆₄

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 185
transitive degree	μ_t(G) = 7744

A minimal faithful permutation representation — 2 generators on 185 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);(2,64)(3,63)(4,62)(5,61)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,53)(14,52)(15,51)(16,50)(17,49)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(25,41)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(66,185)(67,184)(68,183)(69,182)(70,181)(71,180)(72,179)(73,178)(74,177)(75,176)(76,175)(77,174)(78,173)(79,172)(80,171)(81,170)(82,169)(83,168)(84,167)(85,166)(86,165)(87,164)(88,163)(89,162)(90,161)(91,160)(92,159)(93,158)(94,157)(95,156)(96,155)(97,154)(98,153)(99,152)(100,151)(101,150)(102,149)(103,148)(104,147)(105,146)(106,145)(107,144)(108,143)(109,142)(110,141)(111,140)(112,139)(113,138)(114,137)(115,136)(116,135)(117,134)(118,133)(119,132)(120,131)(121,130)(122,129)(123,128)(124,127)(125,126)

Polycyclic presentation:

2^7,11^2|b^2=cd,c^2=de,d^2=ef,e^2=fg,f^2=g,h^11=i,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,h^a=h^10i^10,i^a=i^10

Conjugacy classes & character table

3875 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₅₄₈₈, D₂₃₂₃₂
maximal normal subgroup of	D₁₅₄₈₈
maximal quotient of	Dic₇₇₄₄, D₁₅₄₈₈, D₂₃₂₃₂

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 11, 11]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6*F.7);;
gap> SetPower(coll, 6, F.7);;
gap> SetPower(coll, 8, F.9);;
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, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^10*F.9^10);;
gap> SetConjugate(coll, 9, 1, F.9^10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D7744 #72212907