D₁₁₁₂₈ #72215819

order	22256
automorphism	Aut(G) order 56619264
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₁₂₈

Splittings

Fitting	C₈×C₁₃×C₁₀₇ ⋊ C₂
Hall	C₁₃×C₁₀₇ ⋊ D₈
Sylow	C₁₀₇ ⋊ D₁₀₄
Sylow	C₁₃ ⋊ D₈₅₆

Subgroups

	subgroup	G/H
center	C₂	D₅₅₆₄
commutator, cosocle	C₄×C₁₃×C₁₀₇	C₂²
Frattini	C₄	C₂×D₁₃₉₁
Fitting	C₈×C₁₃×C₁₀₇	C₂
socle	C₂×C₁₃×C₁₀₇	D₄
hypercenter	C₈	D₁₃₉₁
hypocenter	C₁₃×C₁₀₇	D₈
LCS	G ▷ C₄×C₁₃×C₁₀₇ ▷ C₂×C₁₃×C₁₀₇ ▷ C₁₃×C₁₀₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 11128

A minimal faithful permutation representation — 2 generators on 128 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);(2,8)(3,7)(4,6)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(23,128)(24,127)(25,126)(26,125)(27,124)(28,123)(29,122)(30,121)(31,120)(32,119)(33,118)(34,117)(35,116)(36,115)(37,114)(38,113)(39,112)(40,111)(41,110)(42,109)(43,108)(44,107)(45,106)(46,105)(47,104)(48,103)(49,102)(50,101)(51,100)(52,99)(53,98)(54,97)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)

Polycyclic presentation:

2^4,13,107|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^12,f^a=f^106

Conjugacy classes & character table

5567 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₂₂₅₆
maximal normal subgroup of	D₂₂₂₅₆
maximal quotient of	Dic₁₁₁₂₈, D₂₂₂₅₆

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, 13, 107]);;
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.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^106);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11128 #72215819