D₃₅₆₄ #72211151

order	7128
automorphism	Aut(G) order 3849120
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₅₆₄

Splittings

Fitting	C₄×C₁₁×C₈₁ ⋊ C₂
Hall	C₁₁×C₈₁ ⋊ D₄
Sylow	C₈₁ ⋊ D₄₄
Sylow	C₁₁ ⋊ D₃₂₄

Subgroups

	subgroup	G/H
center	C₂	C₂×D₈₉₁
commutator, cosocle	C₂×C₁₁×C₈₁	C₂²
Frattini	C₂×C₂₇	C₂×D₃₃
Fitting	C₄×C₁₁×C₈₁	C₂
socle	C₂×C₃×C₁₁	C₂×D₂₇
hypercenter	C₄	D₈₉₁
hypocenter	C₁₁×C₈₁	D₄
LCS	G ▷ C₂×C₁₁×C₈₁ ▷ C₁₁×C₈₁
UCS	C₁ ◁ C₂ ◁ C₄

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 96
transitive degree	μ_t(G) = 3564

A minimal faithful permutation representation — 2 generators on 96 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)(86,87,88,89,90,91,92,93,94,95,96);(2,4)(6,85)(7,84)(8,83)(9,82)(10,81)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(87,96)(88,95)(89,94)(90,93)(91,92)

Polycyclic presentation:

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

Conjugacy classes & character table

1785 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₇₁₂₈, D₁₀₆₉₂, D₁₇₈₂₀, …
maximal normal subgroup of	D₇₁₂₈
maximal quotient of	D₇₁₂₈, Dic₃₅₆₄, D₁₀₆₉₂, …

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

D3564 #72211151