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D₇⋊A₄ #4279

order	168
isoclinism	isoclinic class of D₇⋊A₄
automorphism	Aut(G) = A₄×AGL(1,𝔽₇) (order 504) · Inn(G) = G · Out(G) = C₃
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	D₇⋊A₄
unambiguous semidirect product	(C₂²⋊ΓL(1,𝔽₈))⋊C₂

Splittings

Fitting, socle, derived	C₂²×C₇ ⋊ C₂×C₃
Hall	C₂²×D₇ ⋊ C₃
Sylow	C₇ ⋊ C₂×A₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₂²×C₇	C₂×C₃
LCS	G ▷ C₂²×C₇
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 28

A minimal faithful permutation representation — 2 generators on 11 points:

(2,6,5,3,4,7)(9,10,11);(1,2,5,6,4,7,3)(8,11)(9,10)

Polycyclic presentation:

2,3,2^2,7|e^a=e^6,c^b=d,d^b=cd,e^b=e^4

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁	C₁₂
size	1	3	6	6	6	6	7	21	28	28	28	28
order	1	2	7	14	14	14	2	2	3	3	6	6
χ₁	1	1	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	1	1	1	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₃	1	1	1	1	1	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$
χ₄	1	1	1	1	1	1	-1	-1	1	1	-1	-1
χ₅	1	1	1	1	1	1	-1	-1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$
χ₆	1	1	1	1	1	1	-1	-1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$
χ₇	3	-1	3	-1	-1	-1	3	-1	0	0	0	0
χ₈	3	-1	3	-1	-1	-1	-3	1	0	0	0	0
χ₉	6	6	-1	-1	-1	-1	0	0	0	0	0	0
χ₁₀	6	-2	-1	−ζ+ζ²+ζ⁴+ζ⁵−ζ⁸+ζ¹⁰+ζ¹¹−ζ¹³+ζ¹⁶+ζ¹⁷+ζ¹⁹−ζ²⁰	ζ+ζ²−ζ⁴+ζ⁵+ζ⁸−ζ¹⁰−ζ¹¹+ζ¹³+ζ¹⁶−ζ¹⁷+ζ¹⁹+ζ²⁰	ζ−ζ²+ζ⁴−ζ⁵+ζ⁸+ζ¹⁰+ζ¹¹+ζ¹³−ζ¹⁶+ζ¹⁷−ζ¹⁹+ζ²⁰	0	0	0	0	0	0
χ₁₁	6	-2	-1	ζ+ζ²−ζ⁴+ζ⁵+ζ⁸−ζ¹⁰−ζ¹¹+ζ¹³+ζ¹⁶−ζ¹⁷+ζ¹⁹+ζ²⁰	ζ−ζ²+ζ⁴−ζ⁵+ζ⁸+ζ¹⁰+ζ¹¹+ζ¹³−ζ¹⁶+ζ¹⁷−ζ¹⁹+ζ²⁰	−ζ+ζ²+ζ⁴+ζ⁵−ζ⁸+ζ¹⁰+ζ¹¹−ζ¹³+ζ¹⁶+ζ¹⁷+ζ¹⁹−ζ²⁰	0	0	0	0	0	0
χ₁₂	6	-2	-1	ζ−ζ²+ζ⁴−ζ⁵+ζ⁸+ζ¹⁰+ζ¹¹+ζ¹³−ζ¹⁶+ζ¹⁷−ζ¹⁹+ζ²⁰	−ζ+ζ²+ζ⁴+ζ⁵−ζ⁸+ζ¹⁰+ζ¹¹−ζ¹³+ζ¹⁶+ζ¹⁷+ζ¹⁹−ζ²⁰	ζ+ζ²−ζ⁴+ζ⁵+ζ⁸−ζ¹⁰−ζ¹¹+ζ¹³+ζ¹⁶−ζ¹⁷+ζ¹⁹+ζ²⁰	0	0	0	0	0	0

Here ζ is a primitive root of unity of order 21 (ζ²¹ = 1); i = ζ⁵.

Relationships

maximal subgroup of	C₂×D₇⋊A₄, A₄×AGL(1,𝔽₇), C₃×D₇⋊A₄, …
maximal normal subgroup of	C₂×D₇⋊A₄, A₄×AGL(1,𝔽₇), C₃×D₇⋊A₄, …
maximal quotient of	#66665, D₇⋊SL(2,𝔽₃), Dic₇⋊A₄, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(168, 49);