D₅₃₂₈ #72211652

order	10656
automorphism	Aut(G) order 9206784
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₅₃₂₈

Splittings

Fitting	C₉×C₁₆×C₃₇ ⋊ C₂
Hall	C₉×C₃₇ ⋊ D₁₆
Sylow	C₃₇ ⋊ D₁₄₄
Sylow	C₉ ⋊ D₅₉₂

Subgroups

	subgroup	G/H
center	C₂	D₂₆₆₄
commutator, cosocle	C₈×C₉×C₃₇	C₂²
Frattini	C₃×C₈	C₂×D₁₁₁
Fitting	C₉×C₁₆×C₃₇	C₂
socle	C₂×C₃×C₃₇	D₂₄
hypercenter	C₁₆	D₃₃₃
hypocenter	C₉×C₃₇	D₁₆
LCS	G ▷ C₈×C₉×C₃₇ ▷ C₄×C₉×C₃₇ ▷ C₂×C₉×C₃₇ ▷ C₉×C₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 62
transitive degree	μ_t(G) = 5328

A minimal faithful permutation representation — 2 generators on 62 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,25)(19,24)(20,23)(21,22)(27,62)(28,61)(29,60)(30,59)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)

Polycyclic presentation:

2^5,3,37,3|b^2=cd,c^2=de,d^2=e,f^3=h,b^a=bc,c^a=cd,d^a=de,f^a=f^2h^2,g^a=g^36,h^a=h^2

Conjugacy classes & character table

2667 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₀₆₅₆, D₁₅₉₈₄, D₂₆₆₄₀
maximal normal subgroup of	D₁₀₆₅₆
maximal quotient of	D₁₀₆₅₆, Dic₅₃₂₈, D₁₅₉₈₄, …

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, 2, 3, 37, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.8^2);;
gap> SetConjugate(coll, 7, 1, F.7^36);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D5328 #72211652