D₁₀₆₆₄ #72215544

order	21328
automorphism	Aut(G) order 53746560
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₀₆₆₄

Splittings

Fitting	C₈×C₃₁×C₄₃ ⋊ C₂
Hall	C₃₁×C₄₃ ⋊ D₈
Sylow	C₄₃ ⋊ D₂₄₈
Sylow	C₃₁ ⋊ D₃₄₄

Subgroups

	subgroup	G/H
center	C₂	D₅₃₃₂
commutator, cosocle	C₄×C₃₁×C₄₃	C₂²
Frattini	C₄	C₂×D₁₃₃₃
Fitting	C₈×C₃₁×C₄₃	C₂
socle	C₂×C₃₁×C₄₃	D₄
hypercenter	C₈	D₁₃₃₃
hypocenter	C₃₁×C₄₃	D₈
LCS	G ▷ C₄×C₃₁×C₄₃ ▷ C₂×C₃₁×C₄₃ ▷ C₃₁×C₄₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 82
transitive degree	μ_t(G) = 10664

A minimal faithful permutation representation — 2 generators on 82 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);(2,8)(3,7)(4,6)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(41,82)(42,81)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)

Polycyclic presentation:

2^4,31,43|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^30,f^a=f^42

Conjugacy classes & character table

5335 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₁₃₂₈, D₃₁₉₉₂
maximal normal subgroup of	D₂₁₃₂₈
maximal quotient of	Dic₁₀₆₆₄, D₂₁₃₂₈, D₃₁₉₉₂

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, 31, 43]);;
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.5^30);;
gap> SetConjugate(coll, 6, 1, F.6^42);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D10664 #72215544