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M₁₆×AGL(1,𝔽₅) #65726

order	320
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

M₁₆×AGL(1,𝔽₅)

Splittings

Fitting	C₅×M₁₆ ⋊ C₄
Sylow	C₅ ⋊ C₄×M₁₆

Subgroups

	subgroup	G/H
center, Frattini	C₄	C₂²×AGL(1,𝔽₅)
commutator, socle	C₂×C₅	C₂×C₄²
Fitting	C₅×M₁₆	C₄
prime radical	C₂²×D₅	C₂×C₄
cosocle	C₄×D₅	C₂³
hypercenter	M₁₆	AGL(1,𝔽₅)
hypocenter	C₅	C₄×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ M₁₆

142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 40

A minimal faithful permutation representation — 3 generators on 13 points:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#100, #10603924, AGL(1,𝔽₅)×#101, …
maximal normal subgroup of	AGL(1,𝔽₅)×#100, #10603924, AGL(1,𝔽₅)×#101, …
maximal quotient of	#10603861, #10603862, #10603864, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1064);