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C₄×C₂²⋊ΓL(1,𝔽₈) #66702

order	336
isoclinism	isoclinic class of C₂²⋊ΓL(1,𝔽₈)
automorphism	Aut(G) = C₂×A₄×AGL(1,𝔽₇) (order 1008) · Inn(G) = C₂²⋊ΓL(1,𝔽₈) · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₄×C₂²⋊ΓL(1,𝔽₈)
unambiguous semidirect product	(C₂²×C₄)⋊ΓL(1,𝔽₈)
unambiguous semidirect product	(C₄×C₇)⋊A₄

Splittings

Fitting, Hall	C₂²×C₄×C₇ ⋊ C₃
derived	C₂²×C₇ ⋊ C₃×C₄
Sylow	C₂²×C₄ ⋊ ΓL(1,𝔽₈)
Sylow	C₇ ⋊ C₄×A₄

Subgroups

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

38 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 84

A minimal faithful permutation representation — 2 generators on 15 points:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	(C₇⋊C₈)⋊A₄, C₈×C₂²⋊ΓL(1,𝔽₈), Dic₁₄⋊A₄, …
maximal normal subgroup of	(C₇⋊C₈)⋊A₄, C₈×C₂²⋊ΓL(1,𝔽₈), Dic₁₄⋊A₄, …
maximal quotient of	C₄×C₇⋊SL(2,𝔽₃), C₈×C₂²⋊ΓL(1,𝔽₈), #131⋊ΓL(1,𝔽₈), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 171);