Finite Group Databasebeta

← Prev (#67412) Next (#67414) →

C₁₉×AGL(1,𝔽₅) #67413

order	380
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	Aut(G) = C₂×C₉×AGL(1,𝔽₅) (order 360) · Inn(G) = AGL(1,𝔽₅) · Out(G) = C₂×C₉
homology	Abelianization (H₁) = C₄×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

C₁₉×AGL(1,𝔽₅)

Splittings

Fitting, socle, Hall	C₅×C₁₉ ⋊ C₄
derived, Sylow	C₅ ⋊ C₄×C₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁₉	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₄×C₁₉
Frattini	C₁	G
Fitting, socle	C₅×C₁₉	C₄
prime radical	D₅×C₁₉	C₂
cosocle	D₅	C₂×C₁₉
LCS	G ▷ C₅
UCS	C₁ ◁ C₁₉

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 95

A minimal faithful permutation representation — 2 generators on 24 points:

(2,5,4,3)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24);(1,2,3,5,4)

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×D₁₉, C₂×C₁₉×AGL(1,𝔽₅), AGL(1,𝔽₅)×C₁₉⋊C₃, …
maximal normal subgroup of	AGL(1,𝔽₅)×D₁₉, C₂×C₁₉×AGL(1,𝔽₅), AGL(1,𝔽₅)×C₁₉⋊C₃, …
maximal quotient of	C₁₉×#170, C₂×C₁₉×AGL(1,𝔽₅), C₃×C₁₉×AGL(1,𝔽₅), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(380, 5);