D₁₁₇₆₀ #72223959

order	23520
automorphism	Aut(G) order 31610880
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₇₆₀

Splittings

Fitting	C₃×C₅×C₁₆×C₄₉ ⋊ C₂
Hall	C₃×C₅×C₄₉ ⋊ D₁₆
Hall	C₅×C₄₉ ⋊ D₄₈
Hall	C₃×C₄₉ ⋊ D₈₀
Sylow	C₄₉ ⋊ D₂₄₀
Hall	C₃×C₅ ⋊ D₇₈₄
Sylow	C₅ ⋊ D₂₃₅₂
Sylow	C₃ ⋊ D₃₉₂₀

Subgroups

	subgroup	G/H
center	C₂	D₅₈₈₀
commutator, cosocle	C₃×C₅×C₈×C₄₉	C₂²
Frattini	C₇×C₈	C₂×D₁₀₅
Fitting	C₃×C₅×C₁₆×C₄₉	C₂
socle	C₂×C₃×C₅×C₇	D₅₆
hypercenter	C₁₆	D₇₃₅
hypocenter	C₃×C₅×C₄₉	D₁₆
LCS	G ▷ C₃×C₅×C₈×C₄₉ ▷ C₃×C₄×C₅×C₄₉ ▷ C₂×C₃×C₅×C₄₉ ▷ C₃×C₅×C₄₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

168 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 63 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) = 73
transitive degree	μ_t(G) = 11760

A minimal faithful permutation representation — 2 generators on 73 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)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,19)(21,24)(22,23)(26,73)(27,72)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)

Polycyclic presentation:

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

Conjugacy classes & character table

5883 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₃₅₂₀
maximal normal subgroup of	D₂₃₅₂₀
maximal quotient of	Dic₁₁₇₆₀, D₂₃₅₂₀

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, 2, 3, 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, 8, F.9);;
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.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.8^6*F.9^6);;
gap> SetConjugate(coll, 9, 1, F.9^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);