D₂₂₀₄ #72211023

order	4408
automorphism	Aut(G) order 2221632
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₂₀₄

Splittings

Fitting	C₄×C₁₉×C₂₉ ⋊ C₂
Hall	C₁₉×C₂₉ ⋊ D₄
Sylow	C₂₉ ⋊ D₇₆
Sylow	C₁₉ ⋊ D₁₁₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₅₅₁
commutator, socle, cosocle	C₂×C₁₉×C₂₉	C₂²
Fitting	C₄×C₁₉×C₂₉	C₂
hypercenter	C₄	D₅₅₁
hypocenter	C₁₉×C₂₉	D₄
LCS	G ▷ C₂×C₁₉×C₂₉ ▷ C₁₉×C₂₉
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 2204

A minimal faithful permutation representation — 2 generators on 52 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);(2,4)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)

Polycyclic presentation:

2^3,19,29|b^2=c,b^a=bc,d^a=d^18,e^a=e^28

Conjugacy classes & character table

1105 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₄₄₀₈, D₆₆₁₂, D₁₁₀₂₀, …
maximal normal subgroup of	D₄₄₀₈
maximal quotient of	D₄₄₀₈, D₆₆₁₂, Dic₂₂₀₄, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 19, 29]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^18);;
gap> SetConjugate(coll, 5, 1, F.5^28);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D2204 #72211023