D₃₁₁₆₀ #72300152

order	62320
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₁₁₆₀

Splittings

Fitting	C₅×C₈×C₁₉×C₄₁ ⋊ C₂
Hall	C₅×C₁₉×C₄₁ ⋊ D₈
Hall	C₁₉×C₄₁ ⋊ D₄₀
Hall	C₅×C₄₁ ⋊ D₁₅₂
Hall	C₅×C₁₉ ⋊ D₃₂₈
Sylow	C₄₁ ⋊ D₇₆₀
Sylow	C₁₉ ⋊ D₁₆₄₀
Sylow	C₅ ⋊ D₆₂₃₂

Subgroups

	subgroup	G/H
center	C₂	D₁₅₅₈₀
commutator, cosocle	C₄×C₅×C₁₉×C₄₁	C₂²
Frattini	C₄	C₂×D₃₈₉₅
Fitting	C₅×C₈×C₁₉×C₄₁	C₂
socle	C₂×C₅×C₁₉×C₄₁	D₄
hypercenter	C₈	D₃₈₉₅
hypocenter	C₅×C₁₉×C₄₁	D₈
LCS	G ▷ C₄×C₅×C₁₉×C₄₁ ▷ C₂×C₅×C₁₉×C₄₁ ▷ C₅×C₁₉×C₄₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 31160

A minimal faithful permutation representation — 2 generators on 73 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);(2,8)(3,7)(4,6)(10,13)(11,12)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(34,73)(35,72)(36,71)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)

Polycyclic presentation:

2^4,5,19,41|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^4,f^a=f^18,g^a=g^40

Conjugacy classes & character table

15583 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, 19, 41]);;
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.6^18);;
gap> SetConjugate(coll, 7, 1, F.7^40);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D31160 #72300152