D₁₄₀₄₀ #72235667

order	28080
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₄₀₄₀

Splittings

Fitting	C₅×C₈×C₁₃×C₂₇ ⋊ C₂
Hall	C₅×C₁₃×C₂₇ ⋊ D₈
Hall	C₁₃×C₂₇ ⋊ D₄₀
Hall	C₅×C₂₇ ⋊ D₁₀₄
Hall	C₅×C₁₃ ⋊ D₂₁₆
Sylow	C₂₇ ⋊ D₅₂₀
Sylow	C₁₃ ⋊ D₁₀₈₀
Sylow	C₅ ⋊ D₂₈₀₈

Subgroups

	subgroup	G/H
center	C₂	D₇₀₂₀
commutator, cosocle	C₄×C₅×C₁₃×C₂₇	C₂²
Frattini	C₄×C₉	C₂×D₁₉₅
Fitting	C₅×C₈×C₁₃×C₂₇	C₂
socle	C₂×C₃×C₅×C₁₃	D₃₆
hypercenter	C₈	D₁₇₅₅
hypocenter	C₅×C₁₃×C₂₇	D₈
LCS	G ▷ C₄×C₅×C₁₃×C₂₇ ▷ C₂×C₅×C₁₃×C₂₇ ▷ C₅×C₁₃×C₂₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

176 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 14040

A minimal faithful permutation representation — 2 generators on 53 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);(2,8)(3,7)(4,6)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(37,40)(38,39)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)

Polycyclic presentation:

2^4,3,5,13,3^2|b^2=cd,c^2=d,e^3=h,h^3=i,b^a=bc,c^a=cd,e^a=e^2h^2i^2,f^a=f^4,g^a=g^12,h^a=h^2i^2,i^a=i^2

Conjugacy classes & character table

7023 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₈₀₈₀
maximal normal subgroup of	D₂₈₀₈₀
maximal quotient of	Dic₁₄₀₄₀, D₂₈₀₈₀

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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 5, 13, 3, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.8^2*F.9^2);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^12);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.9^2);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);