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C₄₃ #189

order	43
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₃×C₇ (order 42) · Inn(G) = C₁ · Out(G) = C₂×C₃×C₇
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₄₃)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

C₄₃

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) = 43
transitive degree	μ_t(G) = 43

A minimal faithful permutation representation — 1 generator on 43 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)

Conjugacy classes & character table

43 conjugacy classes — view the full character table.

Relationships

center of	D₃×C₄₃, D₅×C₄₃, A₄×C₄₃, …
derived subgroup of	D₄₃, C₄₃⋊C₃, Dic₄₃, …
Frattini subgroup of	C₁₈₄₉, D₁₈₄₉, C₂×C₁₈₄₉, …
Fitting subgroup of	D₄₃, C₄₃⋊C₃, D₄₃⋊C₃, …
socle of	D₄₃, C₄₃⋊C₃, D₄₃⋊C₃, …
maximal subgroup of	D₄₃, C₂×C₄₃, C₄₃⋊C₃, …
maximal normal subgroup of	D₄₃, C₂×C₄₃, C₄₃⋊C₃, …
maximal quotient of	C₂×C₄₃, C₃×C₄₃, C₅×C₄₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(43, 1);