Finite Group Databasebeta

← Prev (#65674) Next (#65676) →

C₄×D₅⋊C₈ #65675

order	320
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

C₄×D₅⋊C₈

Splittings

C₅ ⋊ C₂×C₄×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₄²	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₂×C₄×C₈
Frattini	C₂²	C₂²×AGL(1,𝔽₅)
Fitting	C₄²×C₅	C₄
socle	C₂²×C₅	C₂²×C₄
prime radical	C₂²×D₅	C₂×C₄
cosocle	C₂×Dic₅	C₂³
LCS	G ▷ C₅
UCS	C₁ ◁ C₄²

162 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 160

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

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598587, #10598588, #10598589, …
maximal normal subgroup of	#10598587, #10598588, #10598589, …
maximal quotient of	#10603775, #10603779, C₈×D₅⋊C₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1013);