C₄₃⋊C₇ #64507

order	301
isoclinism	isoclinic class of C₄₃⋊C₇
automorphism	Aut(G) = AGL(1,𝔽₄₃) (order 1806) · Inn(G) = G · Out(G) = C₂×C₃
homology	Abelianization (H₁) = C₇ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₄₃⋊C₇

Splittings

Fitting, socle, derived, Sylow

C₄₃ ⋊ C₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₄₃	C₇
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:

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

Polycyclic presentation:

7,43|b^a=b^41

Conjugacy classes & character table

13 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄₃⋊C₇, C₂×C₄₃⋊C₇, (C₄₃⋊C₃)⋊C₇, …
maximal normal subgroup of	D₄₃⋊C₇, C₂×C₄₃⋊C₇, (C₄₃⋊C₃)⋊C₇, …
maximal quotient of	C₂×C₄₃⋊C₇, C₃×C₄₃⋊C₇, C₅×C₄₃⋊C₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(301, 1);

C43⋊C7 #64507