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C₂×D₅×C₁₆ #65188

order	320
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

C₂×D₅×C₁₆

Splittings

Fitting	C₂×C₅×C₁₆ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂²×C₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₁₆	D₅
commutator, hypocenter	C₅	C₂²×C₁₆
Frattini	C₈	C₂²×D₅
Fitting	C₂×C₅×C₁₆	C₂
socle	C₂²×C₅	C₂×C₈
prime radical	C₂²×D₅	C₈
cosocle	C₅×C₈	C₂³
LCS	G ▷ C₅
UCS	C₁ ◁ C₂×C₁₆

98 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2)(4,5)(6,7);(3,4,6,7,5)(8,9,10,13,11,14,16,19,12,15,17,20,18,21,22,23);(1,2)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598085, #10598654, (C₂×AGL(1,𝔽₅))⋊C₁₆, …
maximal normal subgroup of	#10598085, #10598654, (C₂×AGL(1,𝔽₅))⋊C₁₆, …
maximal quotient of	C₁₆×Dic₁₀, C₄×D₅×C₁₆, #10599845, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 526);