D₂₅₅₈₄ #72288730

order	51168
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₅₅₈₄

Splittings

Fitting	C₃×C₁₃×C₁₆×C₄₁ ⋊ C₂
Hall	C₃×C₁₃×C₄₁ ⋊ D₁₆
Hall	C₁₃×C₄₁ ⋊ D₄₈
Hall	C₃×C₄₁ ⋊ D₂₀₈
Sylow	C₄₁ ⋊ D₆₂₄
Hall	C₃×C₁₃ ⋊ D₆₅₆
Sylow	C₁₃ ⋊ D₁₉₆₈
Sylow	C₃ ⋊ D₈₅₂₈

Subgroups

	subgroup	G/H
center	C₂	D₁₂₇₉₂
commutator, cosocle	C₃×C₈×C₁₃×C₄₁	C₂²
Frattini	C₈	C₂×D₁₅₉₉
Fitting	C₃×C₁₃×C₁₆×C₄₁	C₂
socle	C₂×C₃×C₁₃×C₄₁	D₈
hypercenter	C₁₆	D₁₅₉₉
hypocenter	C₃×C₁₃×C₄₁	D₁₆
LCS	G ▷ C₃×C₈×C₁₃×C₄₁ ▷ C₃×C₄×C₁₃×C₄₁ ▷ C₂×C₃×C₁₃×C₄₁ ▷ C₃×C₁₃×C₄₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 25584

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)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(34,73)(35,72)(36,71)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)

Polycyclic presentation:

2^5,3,13,41|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^40

Conjugacy classes & character table

12795 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 13, 41]);;
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.8^40);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D25584 #72288730