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C₃⋊AGL(1,𝔽₇) #1260

order	126
isoclinism	isoclinic class of C₃⋊AGL(1,𝔽₇)
automorphism	Aut(G) = D₃×AGL(1,𝔽₇) (order 252) · Inn(G) = G · Out(G) = C₂
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	C₃⋊AGL(1,𝔽₇)
unambiguous semidirect product	ΓL(1,𝔽₈)⋊D₃
unambiguous semidirect product	D₂₁⋊C₃

Splittings

Fitting, socle, derived	C₃×C₇ ⋊ C₂×C₃
Hall	C₃×ΓL(1,𝔽₈) ⋊ C₂
Sylow	C₇ ⋊ C₃×D₃

Subgroups

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

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 21

A minimal faithful permutation representation — 2 generators on 10 points:

(2,3)(5,8,9,6,7,10);(1,2,3)(4,5,7,10,9,8,6)

Polycyclic presentation:

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

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁	C₁₂
size	1	2	6	6	6	7	7	14	14	21	21	21
order	1	3	7	21	21	3	3	3	3	2	6	6
χ₁	1	1	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	1	1	1	1	1	1	1	-1	-1	-1
χ₃	1	1	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₄	1	1	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	-1	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$
χ₅	1	1	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$
χ₆	1	1	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	-1	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$
χ₇	2	-1	2	-1	-1	2	2	-1	-1	0	0	0
χ₈	2	-1	2	-1	-1	$- 1 + i \sqrt{3}$	$- 1 - i \sqrt{3}$	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	0	0	0
χ₉	2	-1	2	-1	-1	$- 1 - i \sqrt{3}$	$- 1 + i \sqrt{3}$	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	0	0	0
χ₁₀	6	6	-1	-1	-1	0	0	0	0	0	0	0
χ₁₁	6	-3	-1	$\frac{1 - \sqrt{21}}{2}$	$\frac{1 + \sqrt{21}}{2}$	0	0	0	0	0	0	0
χ₁₂	6	-3	-1	$\frac{1 + \sqrt{21}}{2}$	$\frac{1 - \sqrt{21}}{2}$	0	0	0	0	0	0	0

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

Relationships

maximal subgroup of	D₃×AGL(1,𝔽₇), C₂×C₃⋊AGL(1,𝔽₇), #67365, …
maximal normal subgroup of	D₃×AGL(1,𝔽₇), C₂×C₃⋊AGL(1,𝔽₇), C₃×C₃⋊AGL(1,𝔽₇), …
maximal quotient of	C₃⋊(ΓL(1,𝔽₈)⋊C₄), C₂×C₃⋊AGL(1,𝔽₇), #67365, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(126, 9);