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GL(3,𝔽₂) #4272

order	168
isoclinism	isoclinic class of GL(3,𝔽₂)
automorphism	Aut(G) = SO(3,𝔽₇) (order 336) · Inn(G) = G · Out(G) = C₂
holomorph	G ⋊ Aut(G) = #72244493
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₂

Aliases

general linear	GL(3,𝔽₂)
projective special linear	PSL(2,𝔽₇)
projective special linear	PSL(3,𝔽₂)
special linear	SL(3,𝔽₂)
projective general linear	PGL(3,𝔽₂)
projective symplectic	PSp(2,𝔽₇)
omega	Ω(3,𝔽₇)
projective special unitary	PSU(2,𝔽₇)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	G	C₁
LCS	G
UCS	C₁

15 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:

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

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆
size	1	21	24	24	42	56
order	1	2	7	7	4	3
χ₁	1	1	1	1	1	1
χ₂	3	-1	$\frac{- 1 + i \sqrt{7}}{2}$	$\frac{- 1 - i \sqrt{7}}{2}$	1	0
χ₃	3	-1	$\frac{- 1 - i \sqrt{7}}{2}$	$\frac{- 1 + i \sqrt{7}}{2}$	1	0
χ₄	6	2	-1	-1	0	0
χ₅	7	-1	0	0	-1	1
χ₆	8	0	1	1	0	-1

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

Relationships

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(168, 42);