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C₂×GL(3,𝔽₂) #66740

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

non-solvable

Aliases

C₂×GL(3,𝔽₂)

Subgroups

	subgroup	G/H
center, Fitting, hypercenter	C₂	GL(3,𝔽₂)
commutator, hypocenter	GL(3,𝔽₂)	C₂
Frattini	C₁	G
LCS	G ▷ GL(3,𝔽₂)
UCS	C₁ ◁ C₂

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 9
transitive degree	μ_t(G) = 14

A minimal faithful permutation representation — 2 generators on 9 points:

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

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁	C₁₂
size	1	21	24	24	42	56	1	21	24	24	42	56
order	1	2	7	7	4	3	2	2	14	14	4	6
χ₁	1	1	1	1	1	1	1	1	1	1	1	1
χ₂	3	-1	$\frac{- 1 + i \sqrt{7}}{2}$	$\frac{- 1 - i \sqrt{7}}{2}$	1	0	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	3	-1	$\frac{- 1 - i \sqrt{7}}{2}$	$\frac{- 1 + i \sqrt{7}}{2}$	1	0
χ₄	6	2	-1	-1	0	0	6	2	-1	-1	0	0
χ₅	7	-1	0	0	-1	1	7	-1	0	0	-1	1
χ₆	8	0	1	1	0	-1	8	0	1	1	0	-1
χ₇	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1
χ₈	3	-1	$\frac{- 1 + i \sqrt{7}}{2}$	$\frac{- 1 - i \sqrt{7}}{2}$	1	0	-3	1	$\frac{1 - i \sqrt{7}}{2}$	$\frac{1 + i \sqrt{7}}{2}$	-1	0
χ₉	3	-1	$\frac{- 1 - i \sqrt{7}}{2}$	$\frac{- 1 + i \sqrt{7}}{2}$	1	0	-3	1	$\frac{1 + i \sqrt{7}}{2}$	$\frac{1 - i \sqrt{7}}{2}$	-1	0
χ₁₀	6	2	-1	-1	0	0	-6	-2	1	1	0	0
χ₁₁	7	-1	0	0	-1	1	-7	1	0	0	1	-1
χ₁₂	8	0	1	1	0	-1	-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(336, 209);