Finite Group Databasebeta

← Prev (#314) Next (#316) →

C₂×C₃₁ #315

order	62
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₃×C₅ (order 30) · Inn(G) = C₁ · Out(G) = C₂×C₃×C₅
holomorph	G ⋊ Aut(G) = C₂×AGL(1,𝔽₃₁)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₆₂
abelian	C₂×C₃₁

Subgroups

	subgroup	G/H
center	G	C₁
commutator, Frattini	C₁	G
LCS	G ▷ C₁
UCS	C₁ ◁ G

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 33
transitive degree	μ_t(G) = 62

A minimal faithful permutation representation — 1 generator on 33 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)

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

center of	D₄×C₃₁, Q₈×C₃₁, Dic₃×C₃₁, …
derived subgroup of	Dic₆₂, D₁₂₄, C₂²⋊D₃₁, …
Frattini subgroup of	C₄×C₉₆₁, Dic₉₆₁, C₄×C₅×C₉₆₁, …
Fitting subgroup of	Dic₃₁, C₂×D₃₁, C₂×C₃₁⋊C₃, …
socle of	Dic₃₁, C₄×C₃₁, C₂×D₃₁, …
maximal subgroup of	Dic₃₁, C₄×C₃₁, C₂×D₃₁, …
maximal normal subgroup of	Dic₃₁, C₄×C₃₁, C₂×D₃₁, …
maximal quotient of	C₄×C₃₁, C₂²×C₃₁, D₃×C₃₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(62, 2);