C₄₃⋊C₃ #3597

order	129
isoclinism	isoclinic class of C₄₃⋊C₃
automorphism	Aut(G) = AGL(1,𝔽₄₃) (order 1806) · Inn(G) = G · Out(G) = C₂×C₇
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₄₃⋊C₃

Splittings

Fitting, socle, derived, Sylow

C₄₃ ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₄₃	C₃
LCS	G ▷ C₄₃
UCS	C₁

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 43
transitive degree	μ_t(G) = 43

A minimal faithful permutation representation — 2 generators on 43 points:

(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,33)(18,26,31)(21,29,28)(23,32,25)(27,36,40)(30,37,39)(34,41,35)(38,42,43);(1,2,4,8,15,20,5,9,16,25,35,32,12,19,28,24,34,41,22,31,39,36,14,23,33,40,43,42,26,7,13,21,30,38,37,17,3,6,11,18,27,29,10)

Polycyclic presentation:

3,43|b^a=b^36

Conjugacy classes & character table

17 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄₃⋊C₃, C₂×C₄₃⋊C₃, C₃×C₄₃⋊C₃, …
maximal normal subgroup of	D₄₃⋊C₃, C₂×C₄₃⋊C₃, C₃×C₄₃⋊C₃, …
maximal quotient of	C₂×C₄₃⋊C₃, C₄₃⋊C₉, C₃×C₄₃⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(129, 1);

C43⋊C3 #3597