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D₅×C₄⋊C₄ #4037

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

D₅×C₄⋊C₄

Splittings

Fitting	C₅×C₄⋊C₄ ⋊ C₂
Sylow	C₅ ⋊ C₂×C₄⋊C₄

Subgroups

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

92 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:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#64874, D₅×#102, #65060, …
maximal normal subgroup of	#64874, D₅×#102, #65060, …
maximal quotient of	#64936, #64938, #64939, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 112);