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SL(2,𝔽₇) #66645

order	336
isoclinism	isoclinic class of SL(2,𝔽₇)
automorphism	Aut(G) = SO(3,𝔽₇) (order 336) · Inn(G) = GL(3,𝔽₂) · Out(G) = C₂
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₁

Aliases

special linear	SL(2,𝔽₇)
symplectic	Sp(2,𝔽₇)
special unitary	SU(2,𝔽₇)

Subgroups

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

19 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 16

A minimal faithful permutation representation — 2 generators on 16 points:

(1,14,6,2,12,8,13,7)(3,15,10,5,16,9,4,11);(2,5,6)(3,10,9)(4,15,16)(7,11,13)

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁
size	1	1	24	24	24	24	42	42	42	56	56
order	1	2	7	7	14	14	4	8	8	3	6
χ₁	1	1	1	1	1	1	1	1	1	1	1
χ₂	3	3	$\frac{- 1 + i \sqrt{7}}{2}$	$\frac{- 1 - i \sqrt{7}}{2}$	$\frac{- 1 + i \sqrt{7}}{2}$	$\frac{- 1 - i \sqrt{7}}{2}$	-1	1	1	0	0
χ₃	3	3	$\frac{- 1 - i \sqrt{7}}{2}$	$\frac{- 1 + i \sqrt{7}}{2}$	$\frac{- 1 - i \sqrt{7}}{2}$	$\frac{- 1 + i \sqrt{7}}{2}$	-1	1	1	0	0
χ₄	4	-4	$\frac{1 - i \sqrt{7}}{2}$	$\frac{1 + i \sqrt{7}}{2}$	$\frac{- 1 + i \sqrt{7}}{2}$	$\frac{- 1 - i \sqrt{7}}{2}$	0	0	0	1	-1
χ₅	4	-4	$\frac{1 + i \sqrt{7}}{2}$	$\frac{1 - i \sqrt{7}}{2}$	$\frac{- 1 - i \sqrt{7}}{2}$	$\frac{- 1 + i \sqrt{7}}{2}$	0	0	0	1	-1
χ₆	6	6	-1	-1	-1	-1	2	0	0	0	0
χ₇	6	-6	-1	-1	1	1	0	$\sqrt{2}$	$- \sqrt{2}$	0	0
χ₈	6	-6	-1	-1	1	1	0	$- \sqrt{2}$	$\sqrt{2}$	0	0
χ₉	7	7	0	0	0	0	-1	-1	-1	1	1
χ₁₀	8	8	1	1	1	1	0	0	0	-1	-1
χ₁₁	8	-8	1	1	-1	-1	0	0	0	-1	1

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

Relationships

derived subgroup of	#10621462, #10621463, #10621465, …
maximal subgroup of	#10621462, #10621463, #10621465, …
maximal normal subgroup of	#10621462, #10621463, #10621465, …
maximal quotient of	C₂×SL(2,𝔽₇), C₃×SL(2,𝔽₇), C₅×SL(2,𝔽₇), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 114);