D₂₀₆₃₆ #72274606

order	41272
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₀₆₃₆

Splittings

Fitting	C₄×C₇×C₁₁×C₆₇ ⋊ C₂
Hall	C₇×C₁₁×C₆₇ ⋊ D₄
Hall	C₁₁×C₆₇ ⋊ D₂₈
Hall	C₇×C₆₇ ⋊ D₄₄
Hall	C₇×C₁₁ ⋊ D₂₆₈
Sylow	C₆₇ ⋊ D₃₀₈
Sylow	C₁₁ ⋊ D₁₈₇₆
Sylow	C₇ ⋊ D₂₉₄₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₅₁₅₉
commutator, socle, cosocle	C₂×C₇×C₁₁×C₆₇	C₂²
Fitting	C₄×C₇×C₁₁×C₆₇	C₂
hypercenter	C₄	D₅₁₅₉
hypocenter	C₇×C₁₁×C₆₇	D₄
LCS	G ▷ C₂×C₇×C₁₁×C₆₇ ▷ C₇×C₁₁×C₆₇
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 20636

A minimal faithful permutation representation — 2 generators on 89 points:

(1,2,3,4)(5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(2,4)(6,11)(7,10)(8,9)(13,22)(14,21)(15,20)(16,19)(17,18)(24,89)(25,88)(26,87)(27,86)(28,85)(29,84)(30,83)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)

Polycyclic presentation:

2^3,7,11,67|b^2=c,b^a=bc,d^a=d^6,e^a=e^10,f^a=f^66

Conjugacy classes & character table

10321 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 7, 11, 67]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^6);;
gap> SetConjugate(coll, 5, 1, F.5^10);;
gap> SetConjugate(coll, 6, 1, F.6^66);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D20636 #72274606