D₉₁₄₄ #72213962

order	18288
automorphism	Aut(G) order 27651456
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₉₁₄₄

Splittings

Fitting	C₈×C₉×C₁₂₇ ⋊ C₂
Hall	C₉×C₁₂₇ ⋊ D₈
Sylow	C₁₂₇ ⋊ D₇₂
Sylow	C₉ ⋊ D₁₀₁₆

Subgroups

	subgroup	G/H
center	C₂	D₄₅₇₂
commutator, cosocle	C₄×C₉×C₁₂₇	C₂²
Frattini	C₃×C₄	C₂×D₃₈₁
Fitting	C₈×C₉×C₁₂₇	C₂
socle	C₂×C₃×C₁₂₇	D₁₂
hypercenter	C₈	D₁₁₄₃
hypocenter	C₉×C₁₂₇	D₈
LCS	G ▷ C₄×C₉×C₁₂₇ ▷ C₂×C₉×C₁₂₇ ▷ C₉×C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 144
transitive degree	μ_t(G) = 9144

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);(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,144)(20,143)(21,142)(22,141)(23,140)(24,139)(25,138)(26,137)(27,136)(28,135)(29,134)(30,133)(31,132)(32,131)(33,130)(34,129)(35,128)(36,127)(37,126)(38,125)(39,124)(40,123)(41,122)(42,121)(43,120)(44,119)(45,118)(46,117)(47,116)(48,115)(49,114)(50,113)(51,112)(52,111)(53,110)(54,109)(55,108)(56,107)(57,106)(58,105)(59,104)(60,103)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82)

Polycyclic presentation:

2^4,3,127,3|b^2=cd,c^2=d,e^3=g,b^a=bc,c^a=cd,e^a=e^2g^2,f^a=f^126,g^a=g^2

Conjugacy classes & character table

4575 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₈₂₈₈, D₂₇₄₃₂
maximal normal subgroup of	D₁₈₂₈₈
maximal quotient of	Dic₉₁₄₄, D₁₈₂₈₈, D₂₇₄₃₂

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 127, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.7^2);;
gap> SetConjugate(coll, 6, 1, F.6^126);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D9144 #72213962