Dic₅₃ #6155

order	212
isoclinism	isoclinic class of D₅₃
automorphism	Aut(G) = C₂×AGL(1,𝔽₅₃) (order 5512) · Inn(G) = D₅₃ · Out(G) = C₂²×C₁₃
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₅₃

Splittings

derived, Sylow

C₅₃ ⋊ C₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₃
commutator, hypocenter	C₅₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅₃	C₂
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 212

A minimal faithful permutation representation — 2 generators on 57 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);(1,3,2,4)(6,57)(7,56)(8,55)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(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:

2^2,53|a^2=b,c^a=c^52

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#88324, Dic₁₀₆, C₄×D₅₃, …
maximal normal subgroup of	#88324, Dic₁₀₆, C₄×D₅₃, …
maximal quotient of	#88322, C₂×Dic₅₃, C₃×Dic₅₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(212, 1);

Dic53 #6155