D₂₃₃₂ #72210381

order	4664
automorphism	Aut(G) order 2425280
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₃₃₂

Splittings

Fitting	C₄×C₁₁×C₅₃ ⋊ C₂
Hall	C₁₁×C₅₃ ⋊ D₄
Sylow	C₅₃ ⋊ D₄₄
Sylow	C₁₁ ⋊ D₂₁₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₅₈₃
commutator, socle, cosocle	C₂×C₁₁×C₅₃	C₂²
Fitting	C₄×C₁₁×C₅₃	C₂
hypercenter	C₄	D₅₈₃
hypocenter	C₁₁×C₅₃	D₄
LCS	G ▷ C₂×C₁₁×C₅₃ ▷ C₁₁×C₅₃
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 2332

A minimal faithful permutation representation — 2 generators on 68 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);(2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,68)(18,67)(19,66)(20,65)(21,64)(22,63)(23,62)(24,61)(25,60)(26,59)(27,58)(28,57)(29,56)(30,55)(31,54)(32,53)(33,52)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)

Polycyclic presentation:

2^3,11,53|b^2=c,b^a=bc,d^a=d^10,e^a=e^52

Conjugacy classes & character table

1169 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₄₆₆₄, D₆₉₉₆, D₁₁₆₆₀, …
maximal normal subgroup of	D₄₆₆₄
maximal quotient of	D₄₆₆₄, D₆₉₉₆, Dic₂₃₃₂, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 11, 53]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^10);;
gap> SetConjugate(coll, 5, 1, F.5^52);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D2332 #72210381