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C₂²⋊ΓL(1,𝔽₈) #752

order	84
isoclinism	isoclinic class of C₂²⋊ΓL(1,𝔽₈)
automorphism	Aut(G) = A₄×AGL(1,𝔽₇) (order 504) · Inn(G) = G · Out(G) = C₂×C₃
holomorph	G ⋊ Aut(G) = #72222333
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	C₂²⋊ΓL(1,𝔽₈)
unambiguous semidirect product	C₇⋊A₄

Splittings

Fitting, socle, derived, Hall	C₂²×C₇ ⋊ C₃
Sylow	C₇ ⋊ A₄
Sylow	C₂² ⋊ ΓL(1,𝔽₈)

Subgroups

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

10 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

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

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

Relationships

maximal subgroup of	D₇⋊A₄, C₂×C₂²⋊ΓL(1,𝔽₈), A₄×ΓL(1,𝔽₈), …
maximal normal subgroup of	D₇⋊A₄, C₂×C₂²⋊ΓL(1,𝔽₈), A₄×ΓL(1,𝔽₈), …
maximal quotient of	C₇⋊SL(2,𝔽₃), C₂×C₂²⋊ΓL(1,𝔽₈), C₂²⋊(C₇⋊C₉), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(84, 11);