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C₄×C₄⋊Dic₅ #65223

order	320
isoclinism	isoclinic class of Dic₁₀
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

C₄×C₄⋊Dic₅

Splittings

C₅ ⋊ C₄×C₄⋊C₄

Subgroups

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

194 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 320

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

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598146, #10598195, #10598206, …
maximal normal subgroup of	#10598146, #10598195, #10598206, …
maximal quotient of	C₄×C₄⋊#168, #10600184, #10600200, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 561);