D₂₂₄₄ #72210341

order	4488
automorphism	Aut(G) order 1436160
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₂₄₄

Splittings

Fitting	C₃×C₄×C₁₁×C₁₇ ⋊ C₂
Hall	C₃×C₁₁×C₁₇ ⋊ D₄
Hall	C₁₁×C₁₇ ⋊ D₁₂
Hall	C₃×C₁₇ ⋊ D₄₄
Hall	C₃×C₁₁ ⋊ D₆₈
Sylow	C₁₇ ⋊ D₁₃₂
Sylow	C₁₁ ⋊ D₂₀₄
Sylow	C₃ ⋊ D₇₄₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₅₆₁
commutator, socle, cosocle	C₂×C₃×C₁₁×C₁₇	C₂²
Fitting	C₃×C₄×C₁₁×C₁₇	C₂
hypercenter	C₄	D₅₆₁
hypocenter	C₃×C₁₁×C₁₇	D₄
LCS	G ▷ C₂×C₃×C₁₁×C₁₇ ▷ C₃×C₁₁×C₁₇
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 2244

A minimal faithful permutation representation — 2 generators on 35 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);(2,4)(6,7)(9,18)(10,17)(11,16)(12,15)(13,14)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)

Polycyclic presentation:

2^3,3,11,17|b^2=c,b^a=bc,d^a=d^2,e^a=e^10,f^a=f^16

Conjugacy classes & character table

1125 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₄₄₈₈, D₆₇₃₂, D₁₁₂₂₀, …
maximal normal subgroup of	D₄₄₈₈
maximal quotient of	D₄₄₈₈, D₆₇₃₂, Dic₂₂₄₄, …

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 11, 17]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^10);;
gap> SetConjugate(coll, 6, 1, F.6^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D2244 #72210341