D₃₂₇₆₀ #72304528

order	65520
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₂₇₆₀

Splittings

Fitting	C₅×C₇×C₈×C₉×C₁₃ ⋊ C₂
Hall	C₅×C₇×C₉×C₁₃ ⋊ D₈
Hall	C₇×C₉×C₁₃ ⋊ D₄₀
Hall	C₅×C₉×C₁₃ ⋊ D₅₆
Hall	C₅×C₇×C₁₃ ⋊ D₇₂
Hall	C₅×C₇×C₉ ⋊ D₁₀₄
Hall	C₉×C₁₃ ⋊ D₂₈₀
Hall	C₇×C₁₃ ⋊ D₃₆₀
Hall	C₅×C₁₃ ⋊ D₅₀₄
Hall	C₇×C₉ ⋊ D₅₂₀
Hall	C₅×C₉ ⋊ D₇₂₈
Hall	C₅×C₇ ⋊ D₉₃₆
Sylow	C₁₃ ⋊ D₂₅₂₀
Sylow	C₉ ⋊ D₃₆₄₀
Sylow	C₇ ⋊ D₄₆₈₀
Sylow	C₅ ⋊ D₆₅₅₂

Subgroups

	subgroup	G/H
center	C₂	D₁₆₃₈₀
commutator, cosocle	C₄×C₅×C₇×C₉×C₁₃	C₂²
Frattini	C₃×C₄	C₂×D₁₃₆₅
Fitting	C₅×C₇×C₈×C₉×C₁₃	C₂
socle	C₂×C₃×C₅×C₇×C₁₃	D₁₂
hypercenter	C₈	D₄₀₉₅
hypocenter	C₅×C₇×C₉×C₁₃	D₈
LCS	G ▷ C₄×C₅×C₇×C₉×C₁₃ ▷ C₂×C₅×C₇×C₉×C₁₃ ▷ C₅×C₇×C₉×C₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

264 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 99 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) = 42
transitive degree	μ_t(G) = 32760

A minimal faithful permutation representation — 2 generators on 42 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);(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,22)(20,21)(24,29)(25,28)(26,27)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)

Polycyclic presentation:

2^4,3,5,7,13,3|b^2=cd,c^2=d,e^3=i,b^a=bc,c^a=cd,e^a=e^2i^2,f^a=f^4,g^a=g^6,h^a=h^12,i^a=i^2

Conjugacy classes & character table

16383 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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 5, 7, 13, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, 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.9^2);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 8, 1, F.8^12);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);