D₁₃₆₄ #72210180

order	2728
automorphism	Aut(G) order 818400
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₃₆₄

Splittings

Fitting	C₄×C₁₁×C₃₁ ⋊ C₂
Hall	C₁₁×C₃₁ ⋊ D₄
Sylow	C₃₁ ⋊ D₄₄
Sylow	C₁₁ ⋊ D₁₂₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₄₁
commutator, socle, cosocle	C₂×C₁₁×C₃₁	C₂²
Fitting	C₄×C₁₁×C₃₁	C₂
hypercenter	C₄	D₃₄₁
hypocenter	C₁₁×C₃₁	D₄
LCS	G ▷ C₂×C₁₁×C₃₁ ▷ C₁₁×C₃₁
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 46
transitive degree	μ_t(G) = 1364

A minimal faithful permutation representation — 2 generators on 46 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);(2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)

Polycyclic presentation:

2^3,11,31|b^2=c,b^a=bc,d^a=d^10,e^a=e^30

Conjugacy classes & character table

685 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₂₇₂₈, D₄₀₉₂, D₆₈₂₀, …
maximal normal subgroup of	D₂₇₂₈
maximal quotient of	D₂₇₂₈, D₄₀₉₂, Dic₁₃₆₄, …

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, 31]);;
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.5^30);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D1364 #72210180