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C₄×C₄⋊AGL(1,𝔽₅) #65687

order	320
isoclinism	isoclinic class of #700
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

C₄×C₄⋊AGL(1,𝔽₅)

Splittings

Fitting	C₄²×C₅ ⋊ C₄
Sylow	C₅ ⋊ C₄×C₄⋊C₄

Subgroups

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

194 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 80

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

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598530, #10598544, #10598585, …
maximal normal subgroup of	#10598530, #10598544, #10598585, …
maximal quotient of	#10603778, #10603801, C₈×C₄⋊AGL(1,𝔽₅), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1025);