D₁₃₃₂ #72210168

order	2664
automorphism	Aut(G) order 575424
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₃₃₂

Splittings

Fitting	C₄×C₉×C₃₇ ⋊ C₂
Hall	C₉×C₃₇ ⋊ D₄
Sylow	C₃₇ ⋊ D₃₆
Sylow	C₉ ⋊ D₁₄₈

Subgroups

	subgroup	G/H
center	C₂	C₂×D₃₃₃
commutator, cosocle	C₂×C₉×C₃₇	C₂²
Frattini	C₂×C₃	C₂×D₁₁₁
Fitting	C₄×C₉×C₃₇	C₂
socle	C₂×C₃×C₃₇	C₂×D₃
hypercenter	C₄	D₃₃₃
hypocenter	C₉×C₃₇	D₄
LCS	G ▷ C₂×C₉×C₃₇ ▷ C₉×C₃₇
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 50
transitive degree	μ_t(G) = 1332

A minimal faithful permutation representation — 2 generators on 50 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);(2,4)(6,13)(7,12)(8,11)(9,10)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)

Polycyclic presentation:

2^3,3,37,3|b^2=c,d^3=f,b^a=bc,d^a=d^2f^2,e^a=e^36,f^a=f^2

Conjugacy classes & character table

669 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₂₆₆₄, D₃₉₉₆, D₆₆₆₀, …
maximal normal subgroup of	D₂₆₆₄
maximal quotient of	D₂₆₆₄, D₃₉₉₆, Dic₁₃₃₂, …

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, 37, 3]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.6^2);;
gap> SetConjugate(coll, 5, 1, F.5^36);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D1332 #72210168