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C₂×C₄×ΓL(1,𝔽₁₆) #91987

order	480
isoclinism	isoclinic class of ΓL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

C₂×C₄×ΓL(1,𝔽₁₆)

Splittings

Fitting	C₂×C₃×C₄×C₅ ⋊ C₄
derived, Hall	C₃×C₅ ⋊ C₂×C₄²
Sylow	C₅ ⋊ C₂×C₄×Dic₃
Sylow	C₃ ⋊ C₂×C₄×AGL(1,𝔽₅)

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄	ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃×C₅	C₂×C₄²
Frattini	C₂	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₂×C₃×C₄×C₅	C₄
socle	C₂²×C₃×C₅	C₂×C₄
prime radical	C₂²×C₃×D₅	C₂²
cosocle	C₂×C₃×D₅	C₂³
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄

216 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11746517, #11746522, #11746524, …
maximal normal subgroup of	#11746517, #11746522, #11746524, …
maximal quotient of	C₄×D₅⋊(C₃⋊C₈), #11752335, C₄×#6758, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 1063);