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AGL(1,𝔽₇) #183

order	42
isoclinism	isoclinic class of AGL(1,𝔽₇)
automorphism	Aut(G) = AGL(1,𝔽₇) (order 42) · Inn(G) = G · Out(G) = C₁
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₇)²
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

affine general linear	AGL(1,𝔽₇)
unambiguous semidirect product	D₇⋊C₃
unambiguous semidirect product	ΓL(1,𝔽₈)⋊C₂

Splittings

Fitting, socle, derived, Sylow	C₇ ⋊ C₂×C₃
Hall	ΓL(1,𝔽₈) ⋊ C₂
Hall	D₇ ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇	C₂×C₃
LCS	G ▷ C₇
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 7
transitive degree	μ_t(G) = 7

A minimal faithful permutation representation — 2 generators on 7 points:

(2,5,6,3,4,7);(1,2,4,7,6,5,3)

Polycyclic presentation:

2,3,7|c^a=c^6,c^b=c^2

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇
size	1	6	7	7	7	7	7
order	1	7	2	3	3	6	6
χ₁	1	1	1	1	1	1	1
χ₂	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₃	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$
χ₄	1	1	-1	1	1	-1	-1
χ₅	1	1	-1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$
χ₆	1	1	-1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$
χ₇	6	-1	0	0	0	0	0

Here i is the imaginary unit (i² = −1).

Relationships

maximal subgroup of	C₂×AGL(1,𝔽₇), C₃×AGL(1,𝔽₇), C₃⋊AGL(1,𝔽₇), …
maximal normal subgroup of	C₂×AGL(1,𝔽₇), C₃×AGL(1,𝔽₇), C₅×AGL(1,𝔽₇), …
maximal quotient of	ΓL(1,𝔽₈)⋊C₄, C₂×AGL(1,𝔽₇), D₇⋊C₉, …
automorphism group of	D₇, ΓL(1,𝔽₈), AGL(1,𝔽₇), …
holomorph of	C₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(42, 1);