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C₇×A₄⋊AGL(1,𝔽₅) #13132923

order	1680
isoclinism	isoclinic class of A₄⋊AGL(1,𝔽₅)
automorphism	?
homology	Abelianization (H₁) = C₄×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₇×A₄⋊AGL(1,𝔽₅)
unambiguous semidirect product	A₄⋊(C₇×AGL(1,𝔽₅))

Splittings

Fitting, socle	C₂²×C₅×C₇ ⋊ Dic₃
derived	C₅×A₄ ⋊ C₄×C₇
Hall	C₅×C₇ ⋊ SL(2,ℤ/4)
Sylow	C₅ ⋊ C₇×SL(2,ℤ/4)

Subgroups

	subgroup	G/H
center, hypercenter	C₇	A₄⋊AGL(1,𝔽₅)
commutator, hypocenter	C₅×A₄	C₄×C₇
Frattini	C₁	G
Fitting, socle	C₂²×C₅×C₇	Dic₃
prime radical	C₇×D₅×A₄	C₂
cosocle	D₅×A₄	C₂×C₇
LCS	G ▷ C₅×A₄
UCS	C₁ ◁ C₇

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 140

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

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	ΓL(1,𝔽₈)×A₄⋊AGL(1,𝔽₅), C₇×AGL(1,𝔽₅)×S₄, #72218839, …
maximal normal subgroup of	ΓL(1,𝔽₈)×A₄⋊AGL(1,𝔽₅), C₇×AGL(1,𝔽₅)×S₄, #72218839, …
maximal quotient of	C₇×#91886

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 943);