Dic₆₁ #6921

order	244
isoclinism	isoclinic class of D₆₁
automorphism	Aut(G) = C₂×AGL(1,𝔽₆₁) (order 7320) · Inn(G) = D₆₁ · Out(G) = C₂²×C₃×C₅
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₆₁

Splittings

derived, Sylow

C₆₁ ⋊ C₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₆₁
commutator, hypocenter	C₆₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₆₁	C₂
LCS	G ▷ C₆₁
UCS	C₁ ◁ C₂

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 65
transitive degree	μ_t(G) = 244

A minimal faithful permutation representation — 2 generators on 65 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);(1,3,2,4)(6,65)(7,64)(8,63)(9,62)(10,61)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)

Polycyclic presentation:

2^2,61|a^2=b,c^a=c^60

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#92420, Dic₁₂₂, C₄×D₆₁, …
maximal normal subgroup of	#92420, Dic₁₂₂, C₄×D₆₁, …
maximal quotient of	#92418, C₂×Dic₆₁, C₃×Dic₆₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(244, 1);

Dic61 #6921