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D₄×Dic₅ #4080

order	160
automorphism	Aut(G) = C₂²×AGL(1,𝔽₅)×#120 (order 2560) · Inn(G) = C₂²×D₅ · Out(G) = C₂³×D₄
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

D₄×Dic₅

Splittings

C₅ ⋊ C₄×D₄

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅
commutator	C₂×C₅	C₂²×C₄
Fitting, prime radical	C₂×C₅×D₄	C₂
socle, cosocle	C₂²×C₅	C₂³
hypercenter	C₂×D₄	D₅
hypocenter	C₅	C₄×D₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×D₄

94 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:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄⋊#170, #65045, #65046, …
maximal normal subgroup of	D₄⋊#170, #65045, #65046, …
maximal quotient of	C₂²⋊C₄×Dic₅, #65239, #65240, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 155);