D₁₆₅₆ #72210301

order	3312
automorphism	Aut(G) order 874368
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₆₅₆

Splittings

Fitting	C₈×C₉×C₂₃ ⋊ C₂
Hall	C₉×C₂₃ ⋊ D₈
Sylow	C₂₃ ⋊ D₇₂
Sylow	C₉ ⋊ D₁₈₄

Subgroups

	subgroup	G/H
center	C₂	D₈₂₈
commutator, cosocle	C₄×C₉×C₂₃	C₂²
Frattini	C₃×C₄	C₂×D₆₉
Fitting	C₈×C₉×C₂₃	C₂
socle	C₂×C₃×C₂₃	D₁₂
hypercenter	C₈	D₂₀₇
hypocenter	C₉×C₂₃	D₈
LCS	G ▷ C₄×C₉×C₂₃ ▷ C₂×C₉×C₂₃ ▷ C₉×C₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 1656

A minimal faithful permutation representation — 2 generators on 40 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);(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)

Polycyclic presentation:

2^4,3,23,3|b^2=cd,c^2=d,e^3=g,b^a=bc,c^a=cd,e^a=e^2g^2,f^a=f^22,g^a=g^2

Conjugacy classes & character table

831 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃₃₁₂, D₄₉₆₈, D₈₂₈₀, …
maximal normal subgroup of	D₃₃₁₂
maximal quotient of	D₃₃₁₂, D₄₉₆₈, Dic₁₆₅₆, …

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 23, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.7^2);;
gap> SetConjugate(coll, 6, 1, F.6^22);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D1656 #72210301