D₄₅₁₂ #72211429

order	9024
automorphism	Aut(G) order 6641664
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₄₅₁₂

Splittings

Fitting	C₃×C₃₂×C₄₇ ⋊ C₂
Hall	C₃×C₄₇ ⋊ D₃₂
Sylow	C₄₇ ⋊ D₉₆
Sylow	C₃ ⋊ D₁₅₀₄

Subgroups

	subgroup	G/H
center	C₂	D₂₂₅₆
commutator, cosocle	C₃×C₁₆×C₄₇	C₂²
Frattini	C₁₆	C₂×D₁₄₁
Fitting	C₃×C₃₂×C₄₇	C₂
socle	C₂×C₃×C₄₇	D₁₆
hypercenter	C₃₂	D₁₄₁
hypocenter	C₃×C₄₇	D₃₂
LCS	G ▷ C₃×C₁₆×C₄₇ ▷ C₃×C₈×C₄₇ ▷ C₃×C₄×C₄₇ ▷ C₂×C₃×C₄₇ ▷ C₃×C₄₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 82
transitive degree	μ_t(G) = 4512

A minimal faithful permutation representation — 2 generators on 82 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)(36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82);(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,35)(37,82)(38,81)(39,80)(40,79)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,72)(48,71)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)

Polycyclic presentation:

2^6,3,47|b^2=cd,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^2,h^a=h^46

Conjugacy classes & character table

2259 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₉₀₂₄, D₁₃₅₃₆, D₂₂₅₆₀, …
maximal normal subgroup of	D₉₀₂₄
maximal quotient of	D₉₀₂₄, Dic₄₅₁₂, D₁₃₅₃₆, …

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, 2, 3, 47]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6);;
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, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^46);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D4512 #72211429