D₂₆₆₆₄ #72290002

order	53328
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₆₆₆₄

Splittings

Fitting	C₃×C₈×C₁₁×C₁₀₁ ⋊ C₂
Hall	C₃×C₁₁×C₁₀₁ ⋊ D₈
Hall	C₁₁×C₁₀₁ ⋊ D₂₄
Hall	C₃×C₁₀₁ ⋊ D₈₈
Sylow	C₁₀₁ ⋊ D₂₆₄
Hall	C₃×C₁₁ ⋊ D₈₀₈
Sylow	C₁₁ ⋊ D₂₄₂₄
Sylow	C₃ ⋊ D₈₈₈₈

Subgroups

	subgroup	G/H
center	C₂	D₁₃₃₃₂
commutator, cosocle	C₃×C₄×C₁₁×C₁₀₁	C₂²
Frattini	C₄	C₂×D₃₃₃₃
Fitting	C₃×C₈×C₁₁×C₁₀₁	C₂
socle	C₂×C₃×C₁₁×C₁₀₁	D₄
hypercenter	C₈	D₃₃₃₃
hypocenter	C₃×C₁₁×C₁₀₁	D₈
LCS	G ▷ C₃×C₄×C₁₁×C₁₀₁ ▷ C₂×C₃×C₁₁×C₁₀₁ ▷ C₃×C₁₁×C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 123
transitive degree	μ_t(G) = 26664

A minimal faithful permutation representation — 2 generators on 123 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);(2,8)(3,7)(4,6)(10,11)(13,22)(14,21)(15,20)(16,19)(17,18)(24,123)(25,122)(26,121)(27,120)(28,119)(29,118)(30,117)(31,116)(32,115)(33,114)(34,113)(35,112)(36,111)(37,110)(38,109)(39,108)(40,107)(41,106)(42,105)(43,104)(44,103)(45,102)(46,101)(47,100)(48,99)(49,98)(50,97)(51,96)(52,95)(53,94)(54,93)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)

Polycyclic presentation:

2^4,3,11,101|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^10,g^a=g^100

Conjugacy classes & character table

13335 conjugacy classes. The character table is too large to display.

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, 11, 101]);;
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.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^10);;
gap> SetConjugate(coll, 7, 1, F.7^100);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26664 #72290002