D₂₀₆₈₀ #72274637

order	41360
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₀₆₈₀

Splittings

Fitting	C₅×C₈×C₁₁×C₄₇ ⋊ C₂
Hall	C₅×C₁₁×C₄₇ ⋊ D₈
Hall	C₁₁×C₄₇ ⋊ D₄₀
Hall	C₅×C₄₇ ⋊ D₈₈
Hall	C₅×C₁₁ ⋊ D₃₇₆
Sylow	C₄₇ ⋊ D₄₄₀
Sylow	C₁₁ ⋊ D₁₈₈₀
Sylow	C₅ ⋊ D₄₁₃₆

Subgroups

	subgroup	G/H
center	C₂	D₁₀₃₄₀
commutator, cosocle	C₄×C₅×C₁₁×C₄₇	C₂²
Frattini	C₄	C₂×D₂₅₈₅
Fitting	C₅×C₈×C₁₁×C₄₇	C₂
socle	C₂×C₅×C₁₁×C₄₇	D₄
hypercenter	C₈	D₂₅₈₅
hypocenter	C₅×C₁₁×C₄₇	D₈
LCS	G ▷ C₄×C₅×C₁₁×C₄₇ ▷ C₂×C₅×C₁₁×C₄₇ ▷ C₅×C₁₁×C₄₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 20680

A minimal faithful permutation representation — 2 generators on 71 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);(2,8)(3,7)(4,6)(10,13)(11,12)(15,24)(16,23)(17,22)(18,21)(19,20)(26,71)(27,70)(28,69)(29,68)(30,67)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)

Polycyclic presentation:

2^4,5,11,47|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^4,f^a=f^10,g^a=g^46

Conjugacy classes & character table

10343 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, 5, 11, 47]);;
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.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^10);;
gap> SetConjugate(coll, 7, 1, F.7^46);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D20680 #72274637