D₃₂₂₄₀ #72301977

order	64480
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₂₂₄₀

Splittings

Fitting	C₅×C₁₃×C₁₆×C₃₁ ⋊ C₂
Hall	C₅×C₁₃×C₃₁ ⋊ D₁₆
Hall	C₁₃×C₃₁ ⋊ D₈₀
Hall	C₅×C₃₁ ⋊ D₂₀₈
Hall	C₅×C₁₃ ⋊ D₄₉₆
Sylow	C₃₁ ⋊ D₁₀₄₀
Sylow	C₁₃ ⋊ D₂₄₈₀
Sylow	C₅ ⋊ D₆₄₄₈

Subgroups

	subgroup	G/H
center	C₂	D₁₆₁₂₀
commutator, cosocle	C₅×C₈×C₁₃×C₃₁	C₂²
Frattini	C₈	C₂×D₂₀₁₅
Fitting	C₅×C₁₃×C₁₆×C₃₁	C₂
socle	C₂×C₅×C₁₃×C₃₁	D₈
hypercenter	C₁₆	D₂₀₁₅
hypocenter	C₅×C₁₃×C₃₁	D₁₆
LCS	G ▷ C₅×C₈×C₁₃×C₃₁ ▷ C₄×C₅×C₁₃×C₃₁ ▷ C₂×C₅×C₁₃×C₃₁ ▷ C₅×C₁₃×C₃₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 65
transitive degree	μ_t(G) = 32240

A minimal faithful permutation representation — 2 generators on 65 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,21)(19,20)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)

Polycyclic presentation:

2^5,5,13,31|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^4,g^a=g^12,h^a=h^30

Conjugacy classes & character table

16123 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", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 5, 13, 31]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
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, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^12);;
gap> SetConjugate(coll, 8, 1, F.8^30);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D32240 #72301977