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C₄×#41×ΓL(1,𝔽₈) #13060678

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

C₄×#41×ΓL(1,𝔽₈)

Splittings

Fitting, Hall	C₄×C₇×#41 ⋊ C₃
Sylow	C₇ ⋊ C₃×C₄×#41

Subgroups

	subgroup	G/H
center	C₄²	C₂²×ΓL(1,𝔽₈)
commutator	C₂×C₇	C₂³×C₃×C₄
Frattini	C₂²	C₂⁴×ΓL(1,𝔽₈)
Fitting	C₄×C₇×#41	C₃
socle, cosocle	C₂²×C₇	C₂⁴×C₃
prime radical	C₂×#41×ΓL(1,𝔽₈)	C₂
hypercenter	C₄×#41	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×C₄×#41
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄² ◁ C₄×#41

620 conjugacy classes of subgroups; only the 375 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 375 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 4 generators on 19 points:

(1,2,3,4)(14,15,16)(17,19,18);(5,6)(7,10)(8,9)(11,12)(13,14,15,17,16,18,19);(6,9)(10,12);(5,7,8,11)(6,10,9,12)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#41×AGL(1,𝔽₅)×ΓL(1,𝔽₈), C₄×ΓL(1,𝔽₈)×AGL(1,ℤ/8), C₄×#6512⋊C₃, …
maximal normal subgroup of	C₄×ΓL(1,𝔽₈)×AGL(1,ℤ/8), C₄×#6512⋊C₃, C₄×#41×AGL(1,𝔽₇), …
maximal quotient of	#41×AGL(1,𝔽₅)×ΓL(1,𝔽₈)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 9967);