D₃₉₂₀ #72211259

order	7840
automorphism	Aut(G) order 5268480
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₉₂₀

Splittings

Fitting	C₅×C₁₆×C₄₉ ⋊ C₂
Hall	C₅×C₄₉ ⋊ D₁₆
Sylow	C₄₉ ⋊ D₈₀
Sylow	C₅ ⋊ D₇₈₄

Subgroups

	subgroup	G/H
center	C₂	D₁₉₆₀
commutator, cosocle	C₅×C₈×C₄₉	C₂²
Frattini	C₇×C₈	C₂×D₃₅
Fitting	C₅×C₁₆×C₄₉	C₂
socle	C₂×C₅×C₇	D₅₆
hypercenter	C₁₆	D₂₄₅
hypocenter	C₅×C₄₉	D₁₆
LCS	G ▷ C₅×C₈×C₄₉ ▷ C₄×C₅×C₄₉ ▷ C₂×C₅×C₄₉ ▷ C₅×C₄₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 70
transitive degree	μ_t(G) = 3920

A minimal faithful permutation representation — 2 generators on 70 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,21)(19,20)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)

Polycyclic presentation:

2^5,5,7^2|b^2=cd,c^2=de,d^2=e,g^7=h,b^a=bc,c^a=cd,d^a=de,f^a=f^4,g^a=g^6h^6,h^a=h^6

Conjugacy classes & character table

1963 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₇₈₄₀, D₁₁₇₆₀, D₁₉₆₀₀, …
maximal normal subgroup of	D₇₈₄₀
maximal quotient of	D₇₈₄₀, Dic₃₉₂₀, D₁₁₇₆₀, …

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

D3920 #72211259