D₁₆₄₄ #72210312

order	3288
automorphism	Aut(G) order 894336
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₆₄₄

Splittings

Fitting	C₃×C₄×C₁₃₇ ⋊ C₂
Hall	C₃×C₁₃₇ ⋊ D₄
Sylow	C₁₃₇ ⋊ D₁₂
Sylow	C₃ ⋊ D₅₄₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₄₁₁
commutator, socle, cosocle	C₂×C₃×C₁₃₇	C₂²
Fitting	C₃×C₄×C₁₃₇	C₂
hypercenter	C₄	D₄₁₁
hypocenter	C₃×C₁₃₇	D₄
LCS	G ▷ C₂×C₃×C₁₃₇ ▷ C₃×C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 144
transitive degree	μ_t(G) = 1644

A minimal faithful permutation representation — 2 generators on 144 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);(2,4)(6,7)(9,144)(10,143)(11,142)(12,141)(13,140)(14,139)(15,138)(16,137)(17,136)(18,135)(19,134)(20,133)(21,132)(22,131)(23,130)(24,129)(25,128)(26,127)(27,126)(28,125)(29,124)(30,123)(31,122)(32,121)(33,120)(34,119)(35,118)(36,117)(37,116)(38,115)(39,114)(40,113)(41,112)(42,111)(43,110)(44,109)(45,108)(46,107)(47,106)(48,105)(49,104)(50,103)(51,102)(52,101)(53,100)(54,99)(55,98)(56,97)(57,96)(58,95)(59,94)(60,93)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)

Polycyclic presentation:

2^3,3,137|b^2=c,b^a=bc,d^a=d^2,e^a=e^136

Conjugacy classes & character table

825 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃₂₈₈, D₄₉₃₂, D₈₂₂₀, …
maximal normal subgroup of	D₃₂₈₈
maximal quotient of	D₃₂₈₈, D₄₉₃₂, Dic₁₆₄₄, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 137]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^136);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D1644 #72210312