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C₄×#704 #65750

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

Aliases

Splittings

C₅ ⋊ C₄×M₁₆

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₄²
Frattini	C₂²	C₂²×AGL(1,𝔽₅)
Fitting	C₂²×C₄×C₅	C₄
socle	C₂²×C₅	C₂²×C₄
prime radical	C₂³×C₅	C₂×C₄
cosocle	C₂×Dic₅	C₂³
hypercenter	C₂²×C₄	AGL(1,𝔽₅)
hypocenter	C₅	C₄×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 17 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598505, #10598510, #10598558, …
maximal normal subgroup of	#10598505, #10598510, #10598558, …
maximal quotient of	C₄×#4002, #10604175, C₄×#4004, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1088);