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C₁₆×AGL(1,𝔽₅) #64843

order	320
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	Aut(G) = C₄×D₄×AGL(1,𝔽₅) (order 640) · Inn(G) = AGL(1,𝔽₅) · Out(G) = C₄×D₄
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₄

Aliases

C₁₆×AGL(1,𝔽₅)

Splittings

Fitting	C₅×C₁₆ ⋊ C₄
derived, Sylow	C₅ ⋊ C₄×C₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₁₆	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₄×C₁₆
Frattini	C₈	C₂×AGL(1,𝔽₅)
Fitting	C₅×C₁₆	C₄
socle	C₂×C₅	C₄×C₈
prime radical	C₂×D₅	C₂×C₈
cosocle	C₈×D₅	C₂²
LCS	G ▷ C₅
UCS	C₁ ◁ C₁₆

58 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 2 generators on 21 points:

(2,3,4,5);(1,2,3,5,4)(6,7,8,11,9,12,14,17,10,13,15,18,16,19,20,21)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×C₃₂, #10598486, C₂×C₁₆×AGL(1,𝔽₅), …
maximal normal subgroup of	AGL(1,𝔽₅)×C₃₂, #10598486, C₂×C₁₆×AGL(1,𝔽₅), …
maximal quotient of	AGL(1,𝔽₅)×C₃₂, #10598486, C₁₆×#170, …
automorphism group of	D₅×C₁₇, #64665, C₁₇×AGL(1,𝔽₅)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 181);