C₅×C₆₁ #64555

order	305
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₃×C₄²×C₅ (order 240) · Inn(G) = C₁ · Out(G) = C₃×C₄²×C₅
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₃₀₅
abelian	C₅×C₆₁

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) = 66
transitive degree	μ_t(G) = 305

A minimal faithful permutation representation — 1 generator on 66 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)

Conjugacy classes & character table

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

Relationships

center of	C₅×D₃×C₆₁, C₅×D₅×C₆₁, C₅×A₄×C₆₁, …
derived subgroup of	D₃₀₅, Dic₃₀₅, #11927412, …
Fitting subgroup of	D₅×C₆₁, C₅×D₆₁, D₃₀₅, …
socle of	D₅×C₆₁, C₅×D₆₁, D₃₀₅, …
maximal subgroup of	D₅×C₆₁, C₅×D₆₁, D₃₀₅, …
maximal normal subgroup of	D₅×C₆₁, C₅×D₆₁, D₃₀₅, …
maximal quotient of	C₂×C₅×C₆₁, C₃×C₅×C₆₁, C₂₅×C₆₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(305, 2);

C5×C61 #64555