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#41×AGL(1,𝔽₅) #66265

order	320
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

#41×AGL(1,𝔽₅)

Splittings

Fitting	C₅×#41 ⋊ C₄
Sylow	C₅ ⋊ C₄×#41

Subgroups

	subgroup	G/H
center	C₄	C₂²×AGL(1,𝔽₅)
commutator, socle	C₂×C₅	C₂³×C₄
Frattini	C₂	C₂³×AGL(1,𝔽₅)
Fitting	C₅×#41	C₄
prime radical	D₅×#41	C₂
cosocle	C₂×D₅	C₂⁴
hypercenter	#41	AGL(1,𝔽₅)
hypocenter	C₅	C₄×#41
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ #41

310 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 13 points:

(2,3,4,5);(1,2,3,5,4)(6,8,9,12)(7,11,10,13);(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)×#104, #10604002, #10604864, …
maximal normal subgroup of	AGL(1,𝔽₅)×#104, #10604002, #10604864, …
maximal quotient of	#10617114, #10617115, #10617116, …
automorphism group of	D₅⋊C₁₆, #3990, C₂×#674, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1603);