Finite Group Databasebeta

← Prev (#4070) Next (#4072) →

C₂×C₄⋊Dic₅ #4071

order	160
isoclinism	isoclinic class of Dic₁₀
automorphism	Aut(G) = AGL(1,𝔽₅)×#347042 (order 10240) · Inn(G) = C₂×D₅ · Out(G) = #6629386
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

C₂×C₄⋊Dic₅

Splittings

C₅ ⋊ C₂×C₄⋊C₄

Subgroups

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

92 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#64749, #64771, #64774, …
maximal normal subgroup of	#64749, #64771, #64774, …
maximal quotient of	C₂×C₄⋊#168, #65213, C₄×C₄⋊Dic₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 146);