D₉₇₆₀ #72214982

order	19520
automorphism	Aut(G) order 37478400
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₉₇₆₀

Splittings

Fitting	C₅×C₃₂×C₆₁ ⋊ C₂
Hall	C₅×C₆₁ ⋊ D₃₂
Sylow	C₆₁ ⋊ D₁₆₀
Sylow	C₅ ⋊ D₁₉₅₂

Subgroups

	subgroup	G/H
center	C₂	D₄₈₈₀
commutator, cosocle	C₅×C₁₆×C₆₁	C₂²
Frattini	C₁₆	C₂×D₃₀₅
Fitting	C₅×C₃₂×C₆₁	C₂
socle	C₂×C₅×C₆₁	D₁₆
hypercenter	C₃₂	D₃₀₅
hypocenter	C₅×C₆₁	D₃₂
LCS	G ▷ C₅×C₁₆×C₆₁ ▷ C₅×C₈×C₆₁ ▷ C₄×C₅×C₆₁ ▷ C₂×C₅×C₆₁ ▷ C₅×C₆₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 98
transitive degree	μ_t(G) = 9760

A minimal faithful permutation representation — 2 generators on 98 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);(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,37)(35,36)(39,98)(40,97)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)

Polycyclic presentation:

2^6,5,61|b^2=cd,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^4,h^a=h^60

Conjugacy classes & character table

4883 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₉₅₂₀, D₂₉₂₈₀
maximal normal subgroup of	D₁₉₅₂₀
maximal quotient of	Dic₉₇₆₀, D₁₉₅₂₀, D₂₉₂₈₀

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", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 5, 61]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.8^60);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D9760 #72214982