D₁₈₂₁₆ #72273469

order	36432
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₈₂₁₆

Splittings

Fitting	C₈×C₉×C₁₁×C₂₃ ⋊ C₂
Hall	C₉×C₁₁×C₂₃ ⋊ D₈
Hall	C₁₁×C₂₃ ⋊ D₇₂
Hall	C₉×C₂₃ ⋊ D₈₈
Hall	C₉×C₁₁ ⋊ D₁₈₄
Sylow	C₂₃ ⋊ D₇₉₂
Sylow	C₁₁ ⋊ D₁₆₅₆
Sylow	C₉ ⋊ D₂₀₂₄

Subgroups

	subgroup	G/H
center	C₂	D₉₁₀₈
commutator, cosocle	C₄×C₉×C₁₁×C₂₃	C₂²
Frattini	C₃×C₄	C₂×D₇₅₉
Fitting	C₈×C₉×C₁₁×C₂₃	C₂
socle	C₂×C₃×C₁₁×C₂₃	D₁₂
hypercenter	C₈	D₂₂₇₇
hypocenter	C₉×C₁₁×C₂₃	D₈
LCS	G ▷ C₄×C₉×C₁₁×C₂₃ ▷ C₂×C₉×C₁₁×C₂₃ ▷ C₉×C₁₁×C₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 18216

A minimal faithful permutation representation — 2 generators on 51 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);(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,28)(20,27)(21,26)(22,25)(23,24)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)

Polycyclic presentation:

2^4,3,11,23,3|b^2=cd,c^2=d,e^3=h,b^a=bc,c^a=cd,e^a=e^2h^2,f^a=f^10,g^a=g^22,h^a=h^2

Conjugacy classes & character table

9111 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, 3, 11, 23, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.8);;
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.8^2);;
gap> SetConjugate(coll, 6, 1, F.6^10);;
gap> SetConjugate(coll, 7, 1, F.7^22);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D18216 #72273469