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C₂³×D₄×ΓL(1,𝔽₈) #13062294

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product	C₂³×D₄×ΓL(1,𝔽₈)
unambiguous semidirect product	C₄⋊(C₂⁴×ΓL(1,𝔽₈))

Splittings

Fitting, Hall	C₂³×C₇×D₄ ⋊ C₃
Sylow	C₇ ⋊ C₂³×C₃×D₄

Subgroups

	subgroup	G/H
center	C₂⁴	C₂²×ΓL(1,𝔽₈)
commutator, cosocle	C₂×C₇	C₂⁵×C₃
Frattini	C₂	C₂⁵×ΓL(1,𝔽₈)
Fitting	C₂³×C₇×D₄	C₃
socle	C₂⁴×C₇	C₂²×C₃
hypercenter	C₂³×D₄	ΓL(1,𝔽₈)
hypocenter	C₇	C₂³×C₃×D₄
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×D₄

2724 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 5 generators on 17 points:

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×ΓL(1,𝔽₈)×#120, C₂³×D₄×AGL(1,𝔽₇), C₂²×#6500⋊C₃, …
maximal normal subgroup of	C₂²×ΓL(1,𝔽₈)×#120, C₂³×D₄×AGL(1,𝔽₇), C₂²×#6500⋊C₃, …
maximal quotient of	C₂²×ΓL(1,𝔽₈)×#120, ΓL(1,𝔽₈)×#3484, C₂²×ΓL(1,𝔽₈)×#127, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 11583);