C₆₁⋊C₃ #4398

order	183
isoclinism	isoclinic class of C₆₁⋊C₃
automorphism	Aut(G) = AGL(1,𝔽₆₁) (order 3660) · Inn(G) = G · Out(G) = C₄×C₅
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₆₁⋊C₃

Splittings

Fitting, socle, derived, Sylow

C₆₁ ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₆₁	C₃
LCS	G ▷ C₆₁
UCS	C₁

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 61

A minimal faithful permutation representation — 2 generators on 61 points:

(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,34)(18,26,31)(21,29,28)(23,32,25)(27,37,44)(30,39,41)(33,42,38)(35,45,36)(40,48,52)(43,50,49)(46,53,47)(51,56,57)(54,58,55)(59,61,60);(1,2,4,8,15,24,35,46,54,59,58,42,20,5,9,16,25,36,37,14,23,33,43,51,53,32,12,19,28,38,47,48,26,7,13,21,30,40,45,22,31,41,49,55,56,39,17,3,6,11,18,27,34,44,52,57,60,61,50,29,10)

Polycyclic presentation:

3,61|b^a=b^47

Conjugacy classes & character table

23 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₆₁⋊C₃, C₂×C₆₁⋊C₃, C₃×C₆₁⋊C₃, …
maximal normal subgroup of	D₆₁⋊C₃, C₂×C₆₁⋊C₃, C₃×C₆₁⋊C₃, …
maximal quotient of	C₂×C₆₁⋊C₃, C₆₁⋊C₉, C₃×C₆₁⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(183, 1);

C61⋊C3 #4398