D₁₁₈₅₆ #72223980

order	23712
automorphism	Aut(G) order 40974336
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₈₅₆

Splittings

Fitting	C₃×C₁₃×C₁₆×C₁₉ ⋊ C₂
Hall	C₃×C₁₃×C₁₉ ⋊ D₁₆
Hall	C₁₃×C₁₉ ⋊ D₄₈
Hall	C₃×C₁₉ ⋊ D₂₀₈
Hall	C₃×C₁₃ ⋊ D₃₀₄
Sylow	C₁₉ ⋊ D₆₂₄
Sylow	C₁₃ ⋊ D₉₁₂
Sylow	C₃ ⋊ D₃₉₅₂

Subgroups

	subgroup	G/H
center	C₂	D₅₉₂₈
commutator, cosocle	C₃×C₈×C₁₃×C₁₉	C₂²
Frattini	C₈	C₂×D₇₄₁
Fitting	C₃×C₁₃×C₁₆×C₁₉	C₂
socle	C₂×C₃×C₁₃×C₁₉	D₈
hypercenter	C₁₆	D₇₄₁
hypocenter	C₃×C₁₃×C₁₉	D₁₆
LCS	G ▷ C₃×C₈×C₁₃×C₁₉ ▷ C₃×C₄×C₁₃×C₁₉ ▷ C₂×C₃×C₁₃×C₁₉ ▷ C₃×C₁₃×C₁₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 11856

A minimal faithful permutation representation — 2 generators on 51 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,19)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)

Polycyclic presentation:

2^5,3,13,19|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^2,g^a=g^12,h^a=h^18

Conjugacy classes & character table

5931 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₃₇₁₂
maximal normal subgroup of	D₂₃₇₁₂
maximal quotient of	Dic₁₁₈₅₆, D₂₃₇₁₂

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, 3, 13, 19]);;
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.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^12);;
gap> SetConjugate(coll, 8, 1, F.8^18);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11856 #72223980