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C₄₇ #198

order	47
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₂₃ (order 46) · Inn(G) = C₁ · Out(G) = C₂×C₂₃
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₄₇)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

C₄₇

Subgroups

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

2 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 47
transitive degree	μ_t(G) = 47

A minimal faithful permutation representation — 1 generator on 47 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)

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

center of	D₃×C₄₇, D₅×C₄₇, A₄×C₄₇, …
derived subgroup of	D₄₇, Dic₄₇, C₂×D₄₇, …
Frattini subgroup of	C₂₂₀₉, C₇×C₂₂₀₉, C₃²×C₂₂₀₉, …
Fitting subgroup of	D₄₇, C₄₇⋊C₂₃, AGL(1,𝔽₄₇), …
socle of	D₄₇, C₄₇⋊C₂₃, AGL(1,𝔽₄₇), …
maximal subgroup of	D₄₇, C₂×C₄₇, C₃×C₄₇, …
maximal normal subgroup of	D₄₇, C₂×C₄₇, C₃×C₄₇, …
maximal quotient of	C₂×C₄₇, C₃×C₄₇, C₅×C₄₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(47, 1);