D₂₀₇₂₀ #72274656

order	41440
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₀₇₂₀

Splittings

Fitting	C₅×C₇×C₁₆×C₃₇ ⋊ C₂
Hall	C₅×C₇×C₃₇ ⋊ D₁₆
Hall	C₇×C₃₇ ⋊ D₈₀
Hall	C₅×C₃₇ ⋊ D₁₁₂
Sylow	C₃₇ ⋊ D₅₆₀
Hall	C₅×C₇ ⋊ D₅₉₂
Sylow	C₇ ⋊ D₂₉₆₀
Sylow	C₅ ⋊ D₄₁₄₄

Subgroups

	subgroup	G/H
center	C₂	D₁₀₃₆₀
commutator, cosocle	C₅×C₇×C₈×C₃₇	C₂²
Frattini	C₈	C₂×D₁₂₉₅
Fitting	C₅×C₇×C₁₆×C₃₇	C₂
socle	C₂×C₅×C₇×C₃₇	D₈
hypercenter	C₁₆	D₁₂₉₅
hypocenter	C₅×C₇×C₃₇	D₁₆
LCS	G ▷ C₅×C₇×C₈×C₃₇ ▷ C₄×C₅×C₇×C₃₇ ▷ C₂×C₅×C₇×C₃₇ ▷ C₅×C₇×C₃₇
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) = 20720

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)(23,28)(24,27)(25,26)(30,65)(31,64)(32,63)(33,62)(34,61)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)

Polycyclic presentation:

2^5,5,7,37|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^6,h^a=h^36

Conjugacy classes & character table

10363 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, 7, 37]);;
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.7^6);;
gap> SetConjugate(coll, 8, 1, F.8^36);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D20720 #72274656