D₂₀₉₀₄ #72274752

order	41808
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₀₉₀₄

Splittings

Fitting	C₃×C₈×C₁₃×C₆₇ ⋊ C₂
Hall	C₃×C₁₃×C₆₇ ⋊ D₈
Hall	C₁₃×C₆₇ ⋊ D₂₄
Hall	C₃×C₆₇ ⋊ D₁₀₄
Sylow	C₆₇ ⋊ D₃₁₂
Hall	C₃×C₁₃ ⋊ D₅₃₆
Sylow	C₁₃ ⋊ D₁₆₀₈
Sylow	C₃ ⋊ D₆₉₆₈

Subgroups

	subgroup	G/H
center	C₂	D₁₀₄₅₂
commutator, cosocle	C₃×C₄×C₁₃×C₆₇	C₂²
Frattini	C₄	C₂×D₂₆₁₃
Fitting	C₃×C₈×C₁₃×C₆₇	C₂
socle	C₂×C₃×C₁₃×C₆₇	D₄
hypercenter	C₈	D₂₆₁₃
hypocenter	C₃×C₁₃×C₆₇	D₈
LCS	G ▷ C₃×C₄×C₁₃×C₆₇ ▷ C₂×C₃×C₁₃×C₆₇ ▷ C₃×C₁₃×C₆₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 91
transitive degree	μ_t(G) = 20904

A minimal faithful permutation representation — 2 generators on 91 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)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91);(2,8)(3,7)(4,6)(10,11)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(26,91)(27,90)(28,89)(29,88)(30,87)(31,86)(32,85)(33,84)(34,83)(35,82)(36,81)(37,80)(38,79)(39,78)(40,77)(41,76)(42,75)(43,74)(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)

Polycyclic presentation:

2^4,3,13,67|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^12,g^a=g^66

Conjugacy classes & character table

10455 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 13, 67]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
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);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> SetConjugate(coll, 7, 1, F.7^66);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D20904 #72274752