D₃₂₂₂₄ #72301968

order	64448
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₂₂₂₄

Splittings

Fitting	C₁₉×C₃₂×C₅₃ ⋊ C₂
Hall	C₁₉×C₅₃ ⋊ D₃₂
Sylow	C₅₃ ⋊ D₆₀₈
Sylow	C₁₉ ⋊ D₁₆₉₆

Subgroups

	subgroup	G/H
center	C₂	D₁₆₁₁₂
commutator, cosocle	C₁₆×C₁₉×C₅₃	C₂²
Frattini	C₁₆	C₂×D₁₀₀₇
Fitting	C₁₉×C₃₂×C₅₃	C₂
socle	C₂×C₁₉×C₅₃	D₁₆
hypercenter	C₃₂	D₁₀₀₇
hypocenter	C₁₉×C₅₃	D₃₂
LCS	G ▷ C₁₆×C₁₉×C₅₃ ▷ C₈×C₁₉×C₅₃ ▷ C₄×C₁₉×C₅₃ ▷ C₂×C₁₉×C₅₃ ▷ C₁₉×C₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 104
transitive degree	μ_t(G) = 32224

A minimal faithful permutation representation — 2 generators on 104 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)(52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104);(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)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)(53,104)(54,103)(55,102)(56,101)(57,100)(58,99)(59,98)(60,97)(61,96)(62,95)(63,94)(64,93)(65,92)(66,91)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)

Polycyclic presentation:

2^6,19,53|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^18,h^a=h^52

Conjugacy classes & character table

16115 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, 2, 19, 53]);;
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.7^18);;
gap> SetConjugate(coll, 8, 1, F.8^52);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D32224 #72301968